Termination w.r.t. Q proof of /tmp/tmpAqbKie/Ex5Folding.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(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), 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(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod

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(plus, app(s, x)), y) → APP(app(plus, x), y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(times, app(s, x)), y) → APP(times, x)
SUM → APP(fold, add)
PROD → APP(fold, mul)
APP(app(times, app(s, x)), y) → APP(app(times, x), y)
APP(app(times, app(s, x)), y) → APP(plus, app(app(times, x), y))
APP(app(plus, app(s, x)), y) → APP(plus, x)
PROD → APP(s, 0)
APP(app(times, app(s, x)), y) → APP(app(plus, app(app(times, x), y)), y)
SUM → APP(app(fold, add), 0)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
APP(app(plus, app(s, x)), y) → APP(s, app(app(plus, x), y))
PROD → APP(app(fold, mul), app(s, 0))
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(f, y), app(app(app(fold, f), x), z))

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod

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(plus, app(s, x)), y) → APP(app(plus, x), y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(times, app(s, x)), y) → APP(times, x)
SUM → APP(fold, add)
PROD → APP(fold, mul)
APP(app(times, app(s, x)), y) → APP(app(times, x), y)
APP(app(times, app(s, x)), y) → APP(plus, app(app(times, x), y))
APP(app(plus, app(s, x)), y) → APP(plus, x)
PROD → APP(s, 0)
APP(app(times, app(s, x)), y) → APP(app(plus, app(app(times, x), y)), y)
SUM → APP(app(fold, add), 0)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
APP(app(plus, app(s, x)), y) → APP(s, app(app(plus, x), y))
PROD → APP(app(fold, mul), app(s, 0))
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(f, y), app(app(app(fold, f), x), z))

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 10 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(plus, app(s, x)), y) → APP(app(plus, x), y)

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod

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

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

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod

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.

sum
prod

↳ 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(plus, app(s, x)), y) → APP(app(plus, x), y)

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

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)

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:

plus1(s(x), y) → plus1(x, y)

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

fold(x0, x1, nil)
fold(x0, x1, cons(x2, x3))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), 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.

fold(x0, x1, nil)
fold(x0, x1, cons(x2, x3))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)

↳ 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:

plus1(s(x), y) → plus1(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:

plus1(s(x), y) → plus1(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(times, app(s, x)), y) → APP(app(times, x), y)

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod

↳ 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(times, app(s, x)), y) → APP(app(times, x), y)

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

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod

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.

sum
prod

↳ 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(times, app(s, x)), y) → APP(app(times, x), y)

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

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)

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:

times1(s(x), y) → times1(x, y)

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

fold(x0, x1, nil)
fold(x0, x1, cons(x2, x3))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), 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.

fold(x0, x1, nil)
fold(x0, x1, cons(x2, x3))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)

↳ 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:

times1(s(x), y) → times1(x, y)

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

times1(s(x), y) → times1(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(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(f, y), app(app(app(fold, f), x), z))

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod

↳ 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(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(f, y), app(app(app(fold, f), x), z))

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod

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.

sum
prod

↳ 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(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(f, y), app(app(app(fold, f), x), z))

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)

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

APP(app(app(fold, x0), x1), app(app(cons, y2), nil)) → APP(app(x0, y2), x1)
APP(app(app(fold, x0), x1), app(app(cons, y2), app(app(cons, x2), x3))) → APP(app(x0, y2), app(app(x0, x2), app(app(app(fold, x0), x1), x3)))

↳ 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(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
APP(app(app(fold, x0), x1), app(app(cons, y2), app(app(cons, x2), x3))) → APP(app(x0, y2), app(app(x0, x2), app(app(app(fold, x0), x1), x3)))
APP(app(app(fold, x0), x1), app(app(cons, y2), nil)) → APP(app(x0, y2), x1)

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)

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

APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), nil)), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), nil))

↳ 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(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
APP(app(app(fold, x0), x1), app(app(cons, y2), app(app(cons, x2), x3))) → APP(app(x0, y2), app(app(x0, x2), app(app(app(fold, x0), x1), x3)))
APP(app(app(fold, x0), x1), app(app(cons, y2), nil)) → APP(app(x0, y2), x1)
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), nil)), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), nil))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), app(app(cons, y_3), y_4)))

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)

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

APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(fold, x0), x1), app(app(cons, x2), app(app(cons, y_2), nil))) → APP(app(app(fold, x0), x1), app(app(cons, y_2), nil))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))) → APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), nil)), y_4))) → APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), nil)), y_4))
APP(app(app(fold, x0), x1), app(app(cons, x2), app(app(cons, y_2), app(app(cons, y_3), y_4)))) → APP(app(app(fold, x0), x1), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(fold, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(fold, x0), x1), app(app(cons, y_2), 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(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(fold, x0), x1), app(app(cons, x2), app(app(cons, y_2), nil))) → APP(app(app(fold, x0), x1), app(app(cons, y_2), nil))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(fold, x0), x1), app(app(cons, y2), app(app(cons, x2), x3))) → APP(app(x0, y2), app(app(x0, x2), app(app(app(fold, x0), x1), x3)))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))) → APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))
APP(app(app(fold, x0), x1), app(app(cons, y2), nil)) → APP(app(x0, y2), x1)
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), nil)), y_4))) → APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), nil)), y_4))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), nil)), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), nil))
APP(app(app(fold, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(fold, x0), x1), app(app(cons, y_2), y_3))
APP(app(app(fold, x0), x1), app(app(cons, x2), app(app(cons, y_2), app(app(cons, y_3), y_4)))) → APP(app(app(fold, x0), x1), app(app(cons, y_2), app(app(cons, y_3), y_4)))

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), 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

                ↳ 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(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(fold, x0), x1), app(app(cons, x2), app(app(cons, y_2), nil))) → APP(app(app(fold, x0), x1), app(app(cons, y_2), nil))
APP(app(app(fold, x0), x1), app(app(cons, y2), app(app(cons, x2), x3))) → APP(app(x0, y2), app(app(x0, x2), app(app(app(fold, x0), x1), x3)))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))) → APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))
APP(app(app(fold, x0), x1), app(app(cons, y2), nil)) → APP(app(x0, y2), x1)
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), nil)), y_4))) → APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), nil)), y_4))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), nil)), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), nil))
APP(app(app(fold, app(app(fold, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(fold, y_0), y_1), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(fold, x0), x1), app(app(cons, x2), app(app(cons, y_2), app(app(cons, y_3), y_4)))) → APP(app(app(fold, x0), x1), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(fold, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(fold, x0), x1), app(app(cons, y_2), y_3))

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)

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(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(f, y), app(app(app(fold, f), x), z))

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
sum
prod

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.

sum
prod

↳ 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(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(f, y), app(app(app(fold, f), x), z))

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)

The set Q consists of the following terms:

app(app(app(fold, x0), x1), nil)
app(app(app(fold, x0), x1), app(app(cons, x2), x3))
app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), 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

                ↳ UsableRulesProof

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ MNOCProof

                          ↳ QDP

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

APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(f, y)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(app(fold, f), x), z)
APP(app(app(fold, f), x), app(app(cons, y), z)) → APP(app(f, y), app(app(app(fold, f), x), z))

The TRS R consists of the following rules:

app(app(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)

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