Termination w.r.t. Q proof of /tmp/tmpxPg67G/Ex5TermProof.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:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

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:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

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

APP(app(mult, app(s, x)), y) → APP(app(add, app(app(mult, x), y)), y)
FACT → APP(rec, mult)
FACT → APP(s, 0)
APP(app(add, app(s, x)), y) → APP(s, app(app(add, x), y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(add, app(s, x)), y) → APP(app(add, x), y)
APP(app(mult, app(s, x)), y) → APP(mult, x)
APP(app(add, app(s, x)), y) → APP(add, x)
APP(app(mult, app(s, x)), y) → APP(add, app(app(mult, x), y))
FACT → APP(app(rec, mult), app(s, 0))

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

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:

APP(app(mult, app(s, x)), y) → APP(app(add, app(app(mult, x), y)), y)
FACT → APP(rec, mult)
FACT → APP(s, 0)
APP(app(add, app(s, x)), y) → APP(s, app(app(add, x), y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))
APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(add, app(s, x)), y) → APP(app(add, x), y)
APP(app(mult, app(s, x)), y) → APP(mult, x)
APP(app(add, app(s, x)), y) → APP(add, x)
APP(app(mult, app(s, x)), y) → APP(add, app(app(mult, x), y))
FACT → APP(app(rec, mult), app(s, 0))

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 8 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

APP(app(add, app(s, x)), y) → APP(app(add, x), y)

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

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

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

APP(app(add, app(s, x)), y) → APP(app(add, x), y)

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

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

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.

fact

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ATransformationProof

              ↳ QDP

              ↳ QDP

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

APP(app(add, app(s, x)), y) → APP(app(add, x), y)

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

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))

We have to consider all minimal (P,Q,R)-chains.
We have applied the A-Transformation [17] to get from an applicative problem to a standard problem.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ATransformationProof

                          ↳ QDP

                            ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

add1(s(x), y) → add1(x, y)

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

add(0, x0)
add(s(x0), x1)
mult(0, x0)
mult(s(x0), x1)
rec(x0, x1, 0)
rec(x0, x1, s(x2))

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.

add(0, x0)
add(s(x0), x1)
mult(0, x0)
mult(s(x0), x1)
rec(x0, x1, 0)
rec(x0, x1, s(x2))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ATransformationProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

add1(s(x), y) → add1(x, y)

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:

add1(s(x), y) → add1(x, y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)

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

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

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.

fact

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ATransformationProof

              ↳ QDP

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

APP(app(mult, app(s, x)), y) → APP(app(mult, x), y)

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

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))

We have to consider all minimal (P,Q,R)-chains.
We have applied the A-Transformation [17] to get from an applicative problem to a standard problem.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ATransformationProof

                          ↳ QDP

                            ↳ QReductionProof

              ↳ QDP

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

mult1(s(x), y) → mult1(x, y)

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

add(0, x0)
add(s(x0), x1)
mult(0, x0)
mult(s(x0), x1)
rec(x0, x1, 0)
rec(x0, x1, s(x2))

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.

add(0, x0)
add(s(x0), x1)
mult(0, x0)
mult(s(x0), x1)
rec(x0, x1, 0)
rec(x0, x1, s(x2))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ATransformationProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

              ↳ QDP

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

mult1(s(x), y) → mult1(x, y)

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

mult1(s(x), y) → mult1(x, y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                ↳ UsableRulesProof

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

APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), 0) → x
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
fact → app(app(rec, mult), app(s, 0))

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                ↳ UsableRulesProof

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

APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
app(app(app(rec, f), x), 0) → x

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

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.

fact

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                ↳ UsableRulesProof

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

APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
app(app(app(rec, f), x), 0) → x

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y)) at position [1] we obtained the following new rules:

APP(app(app(rec, x0), x1), app(s, 0)) → APP(app(x0, app(s, 0)), x1)
APP(app(app(rec, x0), x1), app(s, app(s, x2))) → APP(app(x0, app(s, app(s, x2))), app(app(x0, app(s, x2)), app(app(app(rec, x0), x1), x2)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ ForwardInstantiation

                ↳ UsableRulesProof

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

APP(app(app(rec, x0), x1), app(s, 0)) → APP(app(x0, app(s, 0)), x1)
APP(app(app(rec, x0), x1), app(s, app(s, x2))) → APP(app(x0, app(s, app(s, x2))), app(app(x0, app(s, x2)), app(app(app(rec, x0), x1), x2)))
APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
app(app(app(rec, f), x), 0) → x

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y)) we obtained the following new rules:

APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, x2)) → APP(app(app(rec, y_0), y_1), app(s, x2))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, 0)) → APP(app(app(rec, y_0), y_1), app(s, 0))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, y_2))) → APP(app(app(rec, y_0), y_1), app(s, app(s, y_2)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ ForwardInstantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

                ↳ UsableRulesProof

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

APP(app(app(rec, x0), x1), app(s, 0)) → APP(app(x0, app(s, 0)), x1)
APP(app(app(rec, x0), x1), app(s, app(s, x2))) → APP(app(x0, app(s, app(s, x2))), app(app(x0, app(s, x2)), app(app(app(rec, x0), x1), x2)))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, x2)) → APP(app(app(rec, y_0), y_1), app(s, x2))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, 0)) → APP(app(app(rec, y_0), y_1), app(s, 0))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, y_2))) → APP(app(app(rec, y_0), y_1), app(s, app(s, y_2)))

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
app(app(app(rec, f), x), 0) → x

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y) we obtained the following new rules:

APP(app(app(rec, x0), x1), app(s, app(s, app(s, y_2)))) → APP(app(app(rec, x0), x1), app(s, app(s, y_2)))
APP(app(app(rec, x0), x1), app(s, app(s, y_2))) → APP(app(app(rec, x0), x1), app(s, y_2))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, 0))) → APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, 0))
APP(app(app(rec, x0), x1), app(s, app(s, 0))) → APP(app(app(rec, x0), x1), app(s, 0))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, y_3))) → APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, y_3))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, app(s, y_3)))) → APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, y_3)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ ForwardInstantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

                                  ↳ QDP

                                    ↳ MNOCProof

                ↳ UsableRulesProof

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

APP(app(app(rec, x0), x1), app(s, app(s, app(s, y_2)))) → APP(app(app(rec, x0), x1), app(s, app(s, y_2)))
APP(app(app(rec, x0), x1), app(s, 0)) → APP(app(x0, app(s, 0)), x1)
APP(app(app(rec, x0), x1), app(s, app(s, x2))) → APP(app(x0, app(s, app(s, x2))), app(app(x0, app(s, x2)), app(app(app(rec, x0), x1), x2)))
APP(app(app(rec, x0), x1), app(s, app(s, y_2))) → APP(app(app(rec, x0), x1), app(s, y_2))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, x2)) → APP(app(app(rec, y_0), y_1), app(s, x2))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, 0)) → APP(app(app(rec, y_0), y_1), app(s, 0))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, 0))) → APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, 0))
APP(app(app(rec, x0), x1), app(s, app(s, 0))) → APP(app(app(rec, x0), x1), app(s, 0))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, y_2))) → APP(app(app(rec, y_0), y_1), app(s, app(s, y_2)))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, y_3))) → APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, y_3))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, app(s, y_3)))) → APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, y_3)))

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
app(app(app(rec, f), x), 0) → x

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ ForwardInstantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

                                  ↳ QDP

                                    ↳ MNOCProof

                                      ↳ QDP

                ↳ UsableRulesProof

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

APP(app(app(rec, x0), x1), app(s, 0)) → APP(app(x0, app(s, 0)), x1)
APP(app(app(rec, x0), x1), app(s, app(s, app(s, y_2)))) → APP(app(app(rec, x0), x1), app(s, app(s, y_2)))
APP(app(app(rec, x0), x1), app(s, app(s, x2))) → APP(app(x0, app(s, app(s, x2))), app(app(x0, app(s, x2)), app(app(app(rec, x0), x1), x2)))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, x2)) → APP(app(app(rec, y_0), y_1), app(s, x2))
APP(app(app(rec, x0), x1), app(s, app(s, y_2))) → APP(app(app(rec, x0), x1), app(s, y_2))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, 0)) → APP(app(app(rec, y_0), y_1), app(s, 0))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, 0))) → APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, 0))
APP(app(app(rec, x0), x1), app(s, app(s, 0))) → APP(app(app(rec, x0), x1), app(s, 0))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, y_2))) → APP(app(app(rec, y_0), y_1), app(s, app(s, y_2)))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, y_3))) → APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, y_3))
APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, app(s, y_3)))) → APP(app(app(rec, app(app(rec, y_0), y_1)), x1), app(s, app(s, y_3)))

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
app(app(app(rec, f), x), 0) → x

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
app(app(app(rec, f), x), 0) → x

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))
fact

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.

fact

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ MNOCProof

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

APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
app(app(app(rec, f), x), 0) → x

The set Q consists of the following terms:

app(app(add, 0), x0)
app(app(add, app(s, x0)), x1)
app(app(mult, 0), x0)
app(app(mult, app(s, x0)), x1)
app(app(app(rec, x0), x1), 0)
app(app(app(rec, x0), x1), app(s, x2))

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

                ↳ UsableRulesProof

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ MNOCProof

                          ↳ QDP

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

APP(app(app(rec, f), x), app(s, y)) → APP(f, app(s, y))
APP(app(app(rec, f), x), app(s, y)) → APP(app(app(rec, f), x), y)
APP(app(app(rec, f), x), app(s, y)) → APP(app(f, app(s, y)), app(app(app(rec, f), x), y))

The TRS R consists of the following rules:

app(app(add, 0), y) → y
app(app(add, app(s, x)), y) → app(s, app(app(add, x), y))
app(app(mult, 0), y) → 0
app(app(mult, app(s, x)), y) → app(app(add, app(app(mult, x), y)), y)
app(app(app(rec, f), x), app(s, y)) → app(app(f, app(s, y)), app(app(app(rec, f), x), y))
app(app(app(rec, f), x), 0) → x

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