Termination w.r.t. Q proof of /tmp/tmpeS2Gkx/cade17.trs

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

Q is empty.

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

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

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

The set Q consists of the following terms:

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

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

SQUARE(s(x)) → DOUBLE(x)
SQUARE(s(x)) → PLUS(square(x), double(x))
DOUBLE(s(x)) → DOUBLE(x)
SQUARE(s(x)) → SQUARE(x)
COND(false, x, y) → LOG(x, square(s(s(y))))
LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)
PLUS(n, s(m)) → PLUS(n, m)
LOG(x, s(s(y))) → LE(x, s(s(y)))
COND(false, x, y) → DOUBLE(log(x, square(s(s(y)))))
LE(s(u), s(v)) → LE(u, v)
COND(false, x, y) → SQUARE(s(s(y)))

The TRS R consists of the following rules:

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

The set Q consists of the following terms:

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

SQUARE(s(x)) → DOUBLE(x)
SQUARE(s(x)) → PLUS(square(x), double(x))
DOUBLE(s(x)) → DOUBLE(x)
SQUARE(s(x)) → SQUARE(x)
COND(false, x, y) → LOG(x, square(s(s(y))))
LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)
PLUS(n, s(m)) → PLUS(n, m)
LOG(x, s(s(y))) → LE(x, s(s(y)))
COND(false, x, y) → DOUBLE(log(x, square(s(s(y)))))
LE(s(u), s(v)) → LE(u, v)
COND(false, x, y) → SQUARE(s(s(y)))

The TRS R consists of the following rules:

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

The set Q consists of the following terms:

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 5 SCCs with 5 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

PLUS(n, s(m)) → PLUS(n, m)

The TRS R consists of the following rules:

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

The set Q consists of the following terms:

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

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

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

PLUS(n, s(m)) → PLUS(n, m)

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

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

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(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

PLUS(n, s(m)) → PLUS(n, m)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

PLUS(n, s(m)) → PLUS(n, m)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

DOUBLE(s(x)) → DOUBLE(x)

The TRS R consists of the following rules:

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

The set Q consists of the following terms:

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

DOUBLE(s(x)) → DOUBLE(x)

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

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

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(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

DOUBLE(s(x)) → DOUBLE(x)

From the DPs we obtained the following set of size-change graphs:

DOUBLE(s(x)) → DOUBLE(x)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

SQUARE(s(x)) → SQUARE(x)

The TRS R consists of the following rules:

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

The set Q consists of the following terms:

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

SQUARE(s(x)) → SQUARE(x)

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

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

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(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

SQUARE(s(x)) → SQUARE(x)

From the DPs we obtained the following set of size-change graphs:

SQUARE(s(x)) → SQUARE(x)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

LE(s(u), s(v)) → LE(u, v)

The TRS R consists of the following rules:

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

The set Q consists of the following terms:

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

LE(s(u), s(v)) → LE(u, v)

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

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

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(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

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

LE(s(u), s(v)) → LE(u, v)

From the DPs we obtained the following set of size-change graphs:

LE(s(u), s(v)) → LE(u, v)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)
COND(false, x, y) → LOG(x, square(s(s(y))))

The TRS R consists of the following rules:

log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

The set Q consists of the following terms:

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)
COND(false, x, y) → LOG(x, square(s(s(y))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

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(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

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

COND(false, x, y) → LOG(x, square(s(s(y))))
LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, x, y) → LOG(x, square(s(s(y)))) at position [1] we obtained the following new rules:

COND(false, x, y) → LOG(x, s(plus(square(s(y)), double(s(y)))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

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

COND(false, x, y) → LOG(x, s(plus(square(s(y)), double(s(y)))))
LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, x, y) → LOG(x, s(plus(square(s(y)), double(s(y))))) at position [1,0,0] we obtained the following new rules:

COND(false, x, y) → LOG(x, s(plus(s(plus(square(y), double(y))), double(s(y)))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

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

LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)
COND(false, x, y) → LOG(x, s(plus(s(plus(square(y), double(y))), double(s(y)))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, x, y) → LOG(x, s(plus(s(plus(square(y), double(y))), double(s(y))))) at position [1,0,1] we obtained the following new rules:

COND(false, x, y) → LOG(x, s(plus(s(plus(square(y), double(y))), s(s(double(y))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

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

LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)
COND(false, x, y) → LOG(x, s(plus(s(plus(square(y), double(y))), s(s(double(y))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, x, y) → LOG(x, s(plus(s(plus(square(y), double(y))), s(s(double(y)))))) at position [1,0] we obtained the following new rules:

COND(false, x, y) → LOG(x, s(s(plus(s(plus(square(y), double(y))), s(double(y))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

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

COND(false, x, y) → LOG(x, s(s(plus(s(plus(square(y), double(y))), s(double(y))))))
LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, x, y) → LOG(x, s(s(plus(s(plus(square(y), double(y))), s(double(y)))))) at position [1,0,0] we obtained the following new rules:

COND(false, x, y) → LOG(x, s(s(s(plus(s(plus(square(y), double(y))), double(y))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

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

LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)
COND(false, x, y) → LOG(x, s(s(s(plus(s(plus(square(y), double(y))), double(y))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y) at position [0] we obtained the following new rules:

LOG(s(x0), s(s(y1))) → COND(le(x0, s(y1)), s(x0), y1)
LOG(0, s(s(y1))) → COND(true, 0, y1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

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

LOG(0, s(s(y1))) → COND(true, 0, y1)
LOG(s(x0), s(s(y1))) → COND(le(x0, s(y1)), s(x0), y1)
COND(false, x, y) → LOG(x, s(s(s(plus(s(plus(square(y), double(y))), double(y))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

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

LOG(s(x0), s(s(y1))) → COND(le(x0, s(y1)), s(x0), y1)
COND(false, x, y) → LOG(x, s(s(s(plus(s(plus(square(y), double(y))), double(y))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule LOG(s(x0), s(s(y1))) → COND(le(x0, s(y1)), s(x0), y1) we obtained the following new rules:

LOG(s(x0), s(s(s(y_4)))) → COND(le(x0, s(s(y_4))), s(x0), s(y_4))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

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

LOG(s(x0), s(s(s(y_4)))) → COND(le(x0, s(s(y_4))), s(x0), s(y_4))
COND(false, x, y) → LOG(x, s(s(s(plus(s(plus(square(y), double(y))), double(y))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND(false, x, y) → LOG(x, s(s(s(plus(s(plus(square(y), double(y))), double(y)))))) we obtained the following new rules:

COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(plus(square(s(z1)), double(s(z1)))), double(s(z1)))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

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

LOG(s(x0), s(s(s(y_4)))) → COND(le(x0, s(s(y_4))), s(x0), s(y_4))
COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(plus(square(s(z1)), double(s(z1)))), double(s(z1)))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(plus(square(s(z1)), double(s(z1)))), double(s(z1))))))) at position [1,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(plus(s(plus(square(z1), double(z1))), double(s(z1)))), double(s(z1)))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

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

LOG(s(x0), s(s(s(y_4)))) → COND(le(x0, s(s(y_4))), s(x0), s(y_4))
COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(plus(s(plus(square(z1), double(z1))), double(s(z1)))), double(s(z1)))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(plus(s(plus(square(z1), double(z1))), double(s(z1)))), double(s(z1))))))) at position [1,0,0,0,0,0,1] we obtained the following new rules:

COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(plus(s(plus(square(z1), double(z1))), s(s(double(z1))))), double(s(z1)))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

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

LOG(s(x0), s(s(s(y_4)))) → COND(le(x0, s(s(y_4))), s(x0), s(y_4))
COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(plus(s(plus(square(z1), double(z1))), s(s(double(z1))))), double(s(z1)))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(plus(s(plus(square(z1), double(z1))), s(s(double(z1))))), double(s(z1))))))) at position [1,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(s(plus(s(plus(square(z1), double(z1))), s(double(z1))))), double(s(z1)))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

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

COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(s(plus(s(plus(square(z1), double(z1))), s(double(z1))))), double(s(z1)))))))
LOG(s(x0), s(s(s(y_4)))) → COND(le(x0, s(s(y_4))), s(x0), s(y_4))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(s(plus(s(plus(square(z1), double(z1))), s(double(z1))))), double(s(z1))))))) at position [1,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(s(z1)))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

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

LOG(s(x0), s(s(s(y_4)))) → COND(le(x0, s(s(y_4))), s(x0), s(y_4))
COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(s(z1)))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(s(z1))))))) at position [1,0,0,0,1] we obtained the following new rules:

COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(s(double(z1))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

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

LOG(s(x0), s(s(s(y_4)))) → COND(le(x0, s(s(y_4))), s(x0), s(y_4))
COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(s(double(z1))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(s(double(z1)))))))) at position [1,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(double(z1))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

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

COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(double(z1))))))))
LOG(s(x0), s(s(s(y_4)))) → COND(le(x0, s(s(y_4))), s(x0), s(y_4))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(double(z1)))))))) at position [1,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

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

LOG(s(x0), s(s(s(y_4)))) → COND(le(x0, s(s(y_4))), s(x0), s(y_4))
COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule LOG(s(x0), s(s(s(y_4)))) → COND(le(x0, s(s(y_4))), s(x0), s(y_4)) we obtained the following new rules:

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule COND(false, s(z0), s(z1)) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1)))))))) we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(square(s(s(z1))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

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

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(square(s(s(z1))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))
LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(square(s(s(z1))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(plus(square(s(z1)), double(s(z1)))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

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

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(plus(square(s(z1)), double(s(z1)))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))
LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(plus(square(s(z1)), double(s(z1)))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(plus(s(plus(square(z1), double(z1))), double(s(z1)))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(plus(s(plus(square(z1), double(z1))), double(s(z1)))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(plus(s(plus(square(z1), double(z1))), double(s(z1)))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,0,0,1] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(plus(s(plus(square(z1), double(z1))), s(s(double(z1))))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(plus(s(plus(square(z1), double(z1))), s(s(double(z1))))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(plus(s(plus(square(z1), double(z1))), s(s(double(z1))))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(s(plus(s(plus(square(z1), double(z1))), s(double(z1))))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(s(plus(s(plus(square(z1), double(z1))), s(double(z1))))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(s(plus(s(plus(square(z1), double(z1))), s(double(z1))))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

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

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1))))))))))
LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(s(s(z1))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,1] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(s(double(s(z1)))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(s(double(s(z1)))))), double(s(s(z1))))))), double(s(s(z1))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(s(double(s(z1)))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(double(s(z1)))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

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

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(double(s(z1)))))), double(s(s(z1))))))), double(s(s(z1))))))))))
LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(double(s(z1)))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(s(z1)))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(s(z1)))))), double(s(s(z1))))))), double(s(s(z1))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(s(z1)))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,0,0,1] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(s(double(z1))))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(s(double(z1))))))), double(s(s(z1))))))), double(s(s(z1))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(s(double(z1))))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(double(z1))))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(double(z1))))))), double(s(s(z1))))))), double(s(s(z1))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), s(double(z1))))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(s(s(z1))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

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

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(s(s(z1))))))), double(s(s(z1))))))))))
LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(s(s(z1))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,1] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(s(double(s(z1)))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

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

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(s(double(s(z1)))))))), double(s(s(z1))))))))))
LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(s(double(s(z1)))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(double(s(z1)))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(double(s(z1)))))))), double(s(s(z1))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(double(s(z1)))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(s(z1)))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(s(z1)))))))), double(s(s(z1))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(s(z1)))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,1] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(s(double(z1))))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

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

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(s(double(z1))))))))), double(s(s(z1))))))))))
LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(s(double(z1))))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(double(z1))))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

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

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(double(z1))))))))), double(s(s(z1))))))))))
LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), s(double(z1))))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), double(s(s(z1))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), double(s(s(z1))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), double(s(s(z1)))))))))) at position [1,0,0,0,0,0,1] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(s(double(s(z1)))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

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

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(s(double(s(z1)))))))))))
LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(s(double(s(z1))))))))))) at position [1,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(double(s(z1)))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(double(s(z1)))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(double(s(z1))))))))))) at position [1,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), double(s(z1)))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), double(s(z1)))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), double(s(z1))))))))))) at position [1,0,0,0,0,0,0,0,1] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(s(double(z1))))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(s(double(z1))))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(s(double(z1)))))))))))) at position [1,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(double(z1))))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

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

LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))
COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(double(z1))))))))))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), s(double(z1)))))))))))) at position [1,0,0,0,0,0,0,0,0] we obtained the following new rules:

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), double(z1))))))))))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ MNOCProof

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

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), double(z1))))))))))))
LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))

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

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Rewriting

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ MNOCProof

                                                                                                                                                                                              ↳ QDP

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

COND(false, s(z0), s(s(s(z1)))) → LOG(s(z0), s(s(s(s(s(s(s(s(s(plus(s(s(s(s(s(s(s(plus(s(s(s(s(s(plus(s(s(s(plus(s(plus(square(z1), double(z1))), double(z1))))), double(z1))))))), double(z1))))))))), double(z1))))))))))))
LOG(s(z0), s(s(s(s(s(y_6)))))) → COND(le(z0, s(s(s(s(y_6))))), s(z0), s(s(s(y_6))))

The TRS R consists of the following rules:

square(s(x)) → s(plus(square(x), double(x)))
square(0) → 0
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
le(0, v) → true
le(s(u), s(v)) → le(u, v)
le(s(u), 0) → false

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