Termination w.r.t. Q proof of /tmp/tmpOKsVwZ/010.trs

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

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

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
inc → app(map, app(app(curry, plus), app(s, 0)))
double → app(map, app(app(curry, times), app(s, app(s, 0))))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
inc → app(map, app(app(curry, plus), app(s, 0)))
double → app(map, app(app(curry, times), app(s, 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(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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
inc → app(map, app(app(curry, plus), app(s, 0)))
double → app(map, app(app(curry, times), app(s, app(s, 0))))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
inc
double

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

DOUBLE → APP(s, 0)
INC → APP(map, app(app(curry, plus), app(s, 0)))
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
APP(app(times, app(s, x)), y) → APP(times, x)
APP(app(times, app(s, x)), y) → APP(app(times, x), y)
DOUBLE → APP(app(curry, times), app(s, app(s, 0)))
APP(app(times, app(s, x)), y) → APP(plus, app(app(times, x), y))
INC → APP(curry, plus)
APP(app(plus, app(s, x)), y) → APP(plus, x)
INC → APP(app(curry, plus), app(s, 0))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(times, app(s, x)), y) → APP(app(plus, app(app(times, x), y)), y)
DOUBLE → APP(map, app(app(curry, times), app(s, app(s, 0))))
APP(app(app(curry, g), x), y) → APP(g, x)
INC → APP(s, 0)
APP(app(app(curry, g), x), y) → APP(app(g, x), y)
APP(app(plus, app(s, x)), y) → APP(s, app(app(plus, x), y))
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
DOUBLE → APP(curry, times)
DOUBLE → APP(s, app(s, 0))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))

The TRS R consists of the following rules:

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
inc → app(map, app(app(curry, plus), app(s, 0)))
double → app(map, app(app(curry, times), app(s, app(s, 0))))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
inc
double

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:

DOUBLE → APP(s, 0)
INC → APP(map, app(app(curry, plus), app(s, 0)))
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
APP(app(times, app(s, x)), y) → APP(times, x)
APP(app(times, app(s, x)), y) → APP(app(times, x), y)
DOUBLE → APP(app(curry, times), app(s, app(s, 0)))
APP(app(times, app(s, x)), y) → APP(plus, app(app(times, x), y))
INC → APP(curry, plus)
APP(app(plus, app(s, x)), y) → APP(plus, x)
INC → APP(app(curry, plus), app(s, 0))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(times, app(s, x)), y) → APP(app(plus, app(app(times, x), y)), y)
DOUBLE → APP(map, app(app(curry, times), app(s, app(s, 0))))
APP(app(app(curry, g), x), y) → APP(g, x)
INC → APP(s, 0)
APP(app(app(curry, g), x), y) → APP(app(g, x), y)
APP(app(plus, app(s, x)), y) → APP(s, app(app(plus, x), y))
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
DOUBLE → APP(curry, times)
DOUBLE → APP(s, app(s, 0))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))

The TRS R consists of the following rules:

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
inc → app(map, app(app(curry, plus), app(s, 0)))
double → app(map, app(app(curry, times), app(s, app(s, 0))))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
inc
double

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 16 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(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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
inc → app(map, app(app(curry, plus), app(s, 0)))
double → app(map, app(app(curry, times), app(s, app(s, 0))))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
inc
double

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(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
inc
double

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.

inc
double

↳ 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(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), 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:

plus1(s(x), y) → plus1(x, y)

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

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
curry(x0, x1, x2)
map(x0, nil)
map(x0, cons(x1, 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.

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
curry(x0, x1, x2)
map(x0, nil)
map(x0, cons(x1, 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:

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(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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
inc → app(map, app(app(curry, plus), app(s, 0)))
double → app(map, app(app(curry, times), app(s, app(s, 0))))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
inc
double

↳ 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(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
inc
double

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.

inc
double

↳ 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(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), 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:

times1(s(x), y) → times1(x, y)

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

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
curry(x0, x1, x2)
map(x0, nil)
map(x0, cons(x1, 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.

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
curry(x0, x1, x2)
map(x0, nil)
map(x0, cons(x1, 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:

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

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

APP(app(app(curry, g), x), y) → APP(g, x)
APP(app(app(curry, g), x), y) → APP(app(g, x), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)

The TRS R consists of the following rules:

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
inc → app(map, app(app(curry, plus), app(s, 0)))
double → app(map, app(app(curry, times), app(s, app(s, 0))))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
inc
double

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

APP(app(app(curry, g), x), y) → APP(g, x)
APP(app(app(curry, g), x), y) → APP(app(g, x), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)

The TRS R consists of the following rules:

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
inc
double

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.

inc
double

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ForwardInstantiation

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

APP(app(app(curry, g), x), y) → APP(g, x)
APP(app(app(curry, g), x), y) → APP(app(g, x), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)

The TRS R consists of the following rules:

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))

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

APP(app(app(curry, app(app(curry, y_0), y_1)), x1), x2) → APP(app(app(curry, y_0), y_1), x1)
APP(app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2)), x2) → APP(app(map, y_0), app(app(cons, y_1), y_2))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ForwardInstantiation

                          ↳ QDP

                            ↳ ForwardInstantiation

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

APP(app(app(curry, app(app(curry, y_0), y_1)), x1), x2) → APP(app(app(curry, y_0), y_1), x1)
APP(app(app(curry, g), x), y) → APP(app(g, x), y)
APP(app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2)), x2) → APP(app(map, y_0), app(app(cons, y_1), y_2))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)

The TRS R consists of the following rules:

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))

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

APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), x2)) → APP(app(app(curry, app(app(curry, y_0), y_1)), y_2), x1)
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), x2)) → APP(app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2)), x1)
APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map, y_0), app(app(cons, y_1), y_2))
APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), x2)) → APP(app(app(curry, y_0), y_1), x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ForwardInstantiation

                          ↳ QDP

                            ↳ ForwardInstantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

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

APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), x2)) → APP(app(app(curry, app(app(curry, y_0), y_1)), y_2), x1)
APP(app(app(curry, app(app(curry, y_0), y_1)), x1), x2) → APP(app(app(curry, y_0), y_1), x1)
APP(app(app(curry, g), x), y) → APP(app(g, x), y)
APP(app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2)), x2) → APP(app(map, y_0), app(app(cons, y_1), y_2))
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), x2)) → APP(app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2)), x1)
APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map, y_0), app(app(cons, y_1), y_2))
APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), x2)) → APP(app(app(curry, y_0), y_1), x1)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)

The TRS R consists of the following rules:

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))

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

APP(app(map, app(map, y_0)), app(app(cons, x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, y_3), y_4))
APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), app(app(cons, y_2), y_3))) → APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, y_2), y_3))
APP(app(map, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map, x0), app(app(cons, y_1), y_2))
APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, y_3), y_4))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ForwardInstantiation

                          ↳ QDP

                            ↳ ForwardInstantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

                                  ↳ QDP

                                    ↳ MNOCProof

                                    ↳ QDPOrderProof

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

APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), x2)) → APP(app(app(curry, app(app(curry, y_0), y_1)), y_2), x1)
APP(app(app(curry, app(app(curry, y_0), y_1)), x1), x2) → APP(app(app(curry, y_0), y_1), x1)
APP(app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2)), x2) → APP(app(map, y_0), app(app(cons, y_1), y_2))
APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), app(app(cons, y_2), y_3))) → APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, y_2), y_3))
APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map, y_0), app(app(cons, y_1), y_2))
APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, y_3), y_4))
APP(app(map, app(map, y_0)), app(app(cons, x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(curry, g), x), y) → APP(app(g, x), y)
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, y_3), y_4))
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), x2)) → APP(app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2)), x1)
APP(app(map, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map, x0), app(app(cons, y_1), y_2))
APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), x2)) → APP(app(app(curry, y_0), y_1), x1)

The TRS R consists of the following rules:

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), 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

                        ↳ ForwardInstantiation

                          ↳ QDP

                            ↳ ForwardInstantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

                                  ↳ QDP

                                    ↳ MNOCProof

                                      ↳ QDP

                                    ↳ QDPOrderProof

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

APP(app(map, app(map, y_0)), app(app(cons, x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), x2)) → APP(app(app(curry, app(app(curry, y_0), y_1)), y_2), x1)
APP(app(app(curry, app(app(curry, y_0), y_1)), x1), x2) → APP(app(app(curry, y_0), y_1), x1)
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, y_3), y_4))
APP(app(app(curry, g), x), y) → APP(app(g, x), y)
APP(app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2)), x2) → APP(app(map, y_0), app(app(cons, y_1), y_2))
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), x2)) → APP(app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2)), x1)
APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), app(app(cons, y_2), y_3))) → APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, y_2), y_3))
APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map, y_0), app(app(cons, y_1), y_2))
APP(app(map, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map, x0), app(app(cons, y_1), y_2))
APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), x2)) → APP(app(app(curry, y_0), y_1), x1)
APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, y_3), y_4))

The TRS R consists of the following rules:

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))

Q is empty.
We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), x2)) → APP(app(app(curry, app(app(curry, y_0), y_1)), y_2), x1)
APP(app(app(curry, app(app(curry, y_0), y_1)), x1), x2) → APP(app(app(curry, y_0), y_1), x1)
APP(app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2)), x2) → APP(app(map, y_0), app(app(cons, y_1), y_2))
APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map, y_0), app(app(cons, y_1), y_2))
APP(app(app(curry, g), x), y) → APP(app(g, x), y)
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), x2)) → APP(app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2)), x1)
APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), x2)) → APP(app(app(curry, y_0), y_1), x1)

The remaining pairs can at least be oriented weakly.

APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), app(app(cons, y_2), y_3))) → APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, y_2), y_3))
APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, y_3), y_4))
APP(app(map, app(map, y_0)), app(app(cons, x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, y_3), y_4))
APP(app(map, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map, x0), app(app(cons, y_1), y_2))

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(APP(x₁, x₂)) = x₁
POL(app(x₁, x₂)) = 1 + x₁ + x₁·x₂
POL(cons) = 0
POL(curry) = 1
POL(map) = 1
POL(nil) = 0
POL(plus) = 0
POL(s) = 0
POL(times) = 0

The following usable rules [17] were oriented:

app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
app(app(times, 0), y) → 0
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(plus, 0), y) → y
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(map, f), nil) → nil
app(app(app(curry, g), x), y) → app(app(g, x), y)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ForwardInstantiation

                          ↳ QDP

                            ↳ ForwardInstantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

                                  ↳ QDP

                                    ↳ MNOCProof

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ UsableRulesProof

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

APP(app(map, app(map, y_0)), app(app(cons, x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, y_3), y_4))
APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), app(app(cons, y_2), y_3))) → APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, y_2), y_3))
APP(app(map, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map, x0), app(app(cons, y_1), y_2))
APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, y_3), y_4))

The TRS R consists of the following rules:

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)
app(app(app(curry, g), x), y) → app(app(g, x), y)
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))

The set Q consists of the following terms:

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ ForwardInstantiation

                          ↳ QDP

                            ↳ ForwardInstantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

                                  ↳ QDP

                                    ↳ MNOCProof

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QDPSizeChangeProof

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

APP(app(map, app(map, y_0)), app(app(cons, x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, y_3), y_4))
APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), app(app(cons, y_2), y_3))) → APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, y_2), y_3))
APP(app(map, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map, x0), app(app(cons, y_1), y_2))
APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, y_3), y_4))

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

app(app(plus, 0), x0)
app(app(plus, app(s, x0)), x1)
app(app(times, 0), x0)
app(app(times, app(s, x0)), x1)
app(app(app(curry, x0), x1), x2)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))

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:

APP(app(map, app(map, y_0)), app(app(cons, x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(map, app(map, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(map, y_0)), app(app(cons, y_1), y_2))), app(app(cons, y_3), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(map, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map, x0), app(app(cons, y_1), y_2))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, x1), app(app(cons, y_2), y_3))) → APP(app(map, app(app(curry, y_0), y_1)), app(app(cons, y_2), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, x1), app(app(cons, y_3), y_4))) → APP(app(map, app(app(curry, app(app(curry, y_0), y_1)), y_2)), app(app(cons, y_3), y_4))
The graph contains the following edges 1 >= 1, 2 > 2