Termination w.r.t. Q proof of TRS/higher-order/BirdTreeHeight.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(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

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))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

Q is empty.

The TRS is overlay and locally confluent. By [15] 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(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

The set Q consists of the following terms:

app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

APP(app(maxlist, x), app(app(cons, y), ys)) → APP(maxlist, y)
APP(height, app(app(node, x), xs)) → APP(maxlist, 0)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(maxlist, y), ys)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(le, app(s, x)), app(s, y)) → APP(le, x)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(le, x)
APP(height, app(app(node, x), xs)) → APP(s, app(app(maxlist, 0), app(app(map, height), xs)))
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(if, app(app(le, x), y))
APP(height, app(app(node, x), xs)) → APP(app(map, height), xs)
APP(height, app(app(node, x), xs)) → APP(map, height)
APP(height, app(app(node, x), xs)) → APP(app(maxlist, 0), app(app(map, height), xs))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(le, x), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))

The TRS R consists of the following rules:

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))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

The set Q consists of the following terms:

app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

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

APP(app(maxlist, x), app(app(cons, y), ys)) → APP(maxlist, y)
APP(height, app(app(node, x), xs)) → APP(maxlist, 0)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(maxlist, y), ys)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(le, app(s, x)), app(s, y)) → APP(le, x)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(le, x)
APP(height, app(app(node, x), xs)) → APP(s, app(app(maxlist, 0), app(app(map, height), xs)))
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(if, app(app(le, x), y))
APP(height, app(app(node, x), xs)) → APP(app(map, height), xs)
APP(height, app(app(node, x), xs)) → APP(map, height)
APP(height, app(app(node, x), xs)) → APP(app(maxlist, 0), app(app(map, height), xs))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(le, x), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))

The TRS R consists of the following rules:

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))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

The set Q consists of the following terms:

app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

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

APP(app(maxlist, x), app(app(cons, y), ys)) → APP(maxlist, y)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(height, app(app(node, x), xs)) → APP(maxlist, 0)
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(maxlist, y), ys)
APP(app(le, app(s, x)), app(s, y)) → APP(le, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(le, x)
APP(height, app(app(node, x), xs)) → APP(s, app(app(maxlist, 0), app(app(map, height), xs)))
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(if, app(app(le, x), y))
APP(height, app(app(node, x), xs)) → APP(app(map, height), xs)
APP(height, app(app(node, x), xs)) → APP(map, height)
APP(height, app(app(node, x), xs)) → APP(app(maxlist, 0), app(app(map, height), xs))
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(le, x), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))

The TRS R consists of the following rules:

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))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

The set Q consists of the following terms:

app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 12 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

                  ↳ QDP

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

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

The TRS R consists of the following rules:

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))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

The set Q consists of the following terms:

app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13]. Here, we combined the reduction pair processor with the A-transformation [14] which results in the following intermediate Q-DP Problem.
Q DP problem:
The TRS P consists of the following rules:

LE(s(x), s(y)) → LE(x, y)

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

map(x0, nil)
map(x0, cons(x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
maxlist(x0, cons(x1, x2))
maxlist(x0, nil)
height(node(x0, x1))

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
LE(x1, x2) = LE(x2)
s(x1) = s(x1)

Lexicographic Path Order [19].
Precedence:

s₁ > LE₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

                  ↳ QDP

                  ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

The set Q consists of the following terms:

app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

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

APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(maxlist, y), ys)

The TRS R consists of the following rules:

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))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

The set Q consists of the following terms:

app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))

MAXLIST(x, cons(y, ys)) → MAXLIST(y, ys)

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

map(x0, nil)
map(x0, cons(x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
maxlist(x0, cons(x1, x2))
maxlist(x0, nil)
height(node(x0, x1))

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

The following pairs can be oriented strictly and are deleted.

APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(maxlist, y), ys)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
MAXLIST(x1, x2) = MAXLIST(x2)
cons(x1, x2) = cons(x1, x2)

Lexicographic Path Order [19].
Precedence:

cons₂ > MAXLIST₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

                  ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

The set Q consists of the following terms:

app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

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

APP(height, app(app(node, x), xs)) → APP(app(map, height), xs)
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(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

The set Q consists of the following terms:

app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

APP(height, app(app(node, x), xs)) → APP(app(map, height), xs)
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 remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
APP(x1, x2) = x2
height = height
app(x1, x2) = app(x1, x2)
node = node
map = map
cons = cons

Lexicographic Path Order [19].
Precedence:

height > map > app₂
node > map > app₂
cons > app₂

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

The set Q consists of the following terms:

app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))

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