Termination w.r.t. Q proof of TRS/higher-order/BirdBTreeMember.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(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

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(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

The set Q consists of the following terms:

app(app(lt, app(s, x0)), app(s, x1))
app(app(lt, 0), app(s, x0))
app(app(lt, x0), 0)
app(app(eq, x0), x0)
app(app(eq, app(s, x0)), 0)
app(app(eq, 0), app(s, x0))
app(app(member, x0), null)
app(app(member, x0), app(app(app(fork, x1), x2), x3))

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(member, w), app(app(app(fork, x), y), z)) → APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(if, app(app(eq, w), y))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(eq, w)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), z)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(lt, w)
APP(app(lt, app(s, x)), app(s, y)) → APP(lt, x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(lt, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(if, app(app(lt, w), y))
APP(app(lt, app(s, x)), app(s, y)) → APP(app(lt, x), y)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(if, app(app(eq, w), y)), true)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(eq, w), y)

The TRS R consists of the following rules:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

The set Q consists of the following terms:

app(app(lt, app(s, x0)), app(s, x1))
app(app(lt, 0), app(s, x0))
app(app(lt, x0), 0)
app(app(eq, x0), x0)
app(app(eq, app(s, x0)), 0)
app(app(eq, 0), app(s, x0))
app(app(member, x0), null)
app(app(member, x0), app(app(app(fork, x1), x2), x3))

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(member, w), app(app(app(fork, x), y), z)) → APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(if, app(app(eq, w), y))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(eq, w)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), z)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(lt, w)
APP(app(lt, app(s, x)), app(s, y)) → APP(lt, x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(lt, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(if, app(app(lt, w), y))
APP(app(lt, app(s, x)), app(s, y)) → APP(app(lt, x), y)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(if, app(app(eq, w), y)), true)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(eq, w), y)

The TRS R consists of the following rules:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

The set Q consists of the following terms:

app(app(lt, app(s, x0)), app(s, x1))
app(app(lt, 0), app(s, x0))
app(app(lt, x0), 0)
app(app(eq, x0), x0)
app(app(eq, app(s, x0)), 0)
app(app(eq, 0), app(s, x0))
app(app(member, x0), null)
app(app(member, x0), app(app(app(fork, x1), x2), x3))

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(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(if, app(app(eq, w), y))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(eq, w)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(if, app(app(lt, w), y)), app(app(member, w), x))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), z)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(lt, w)
APP(app(lt, app(s, x)), app(s, y)) → APP(lt, x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(lt, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z))
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(if, app(app(lt, w), y))
APP(app(lt, app(s, x)), app(s, y)) → APP(app(lt, x), y)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(eq, w), y)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(if, app(app(eq, w), y)), true)

The TRS R consists of the following rules:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

The set Q consists of the following terms:

app(app(lt, app(s, x0)), app(s, x1))
app(app(lt, 0), app(s, x0))
app(app(lt, x0), 0)
app(app(eq, x0), x0)
app(app(eq, app(s, x0)), 0)
app(app(eq, 0), app(s, x0))
app(app(member, x0), null)
app(app(member, x0), app(app(app(fork, x1), x2), x3))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 11 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

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

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

The TRS R consists of the following rules:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

The set Q consists of the following terms:

app(app(lt, app(s, x0)), app(s, x1))
app(app(lt, 0), app(s, x0))
app(app(lt, x0), 0)
app(app(eq, x0), x0)
app(app(eq, app(s, x0)), 0)
app(app(eq, 0), app(s, x0))
app(app(member, x0), null)
app(app(member, x0), app(app(app(fork, x1), x2), x3))

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:

LT(s(x), s(y)) → LT(x, y)

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

lt(s(x0), s(x1))
lt(0, s(x0))
lt(x0, 0)
eq(x0, x0)
eq(s(x0), 0)
eq(0, s(x0))
member(x0, null)
member(x0, fork(x1, x2, x3))

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

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [19].
Precedence:

s₁ > LT₁

Status:

s₁: multiset
LT₁: [1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

                  ↳ QDP

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

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

The set Q consists of the following terms:

app(app(lt, app(s, x0)), app(s, x1))
app(app(lt, 0), app(s, x0))
app(app(lt, x0), 0)
app(app(eq, x0), x0)
app(app(eq, app(s, x0)), 0)
app(app(eq, 0), app(s, x0))
app(app(member, x0), null)
app(app(member, x0), app(app(app(fork, x1), x2), x3))

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

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

APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), z)

The TRS R consists of the following rules:

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

The set Q consists of the following terms:

app(app(lt, app(s, x0)), app(s, x1))
app(app(lt, 0), app(s, x0))
app(app(lt, x0), 0)
app(app(eq, x0), x0)
app(app(eq, app(s, x0)), 0)
app(app(eq, 0), app(s, x0))
app(app(member, x0), null)
app(app(member, x0), app(app(app(fork, x1), x2), x3))

MEMBER(w, fork(x, y, z)) → MEMBER(w, x)
MEMBER(w, fork(x, y, z)) → MEMBER(w, z)

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

lt(s(x0), s(x1))
lt(0, s(x0))
lt(x0, 0)
eq(x0, x0)
eq(s(x0), 0)
eq(0, s(x0))
member(x0, null)
member(x0, fork(x1, x2, x3))

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

The following pairs can be oriented strictly and are deleted.

APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), x)
APP(app(member, w), app(app(app(fork, x), y), z)) → APP(app(member, w), z)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
MEMBER(x1, x2) = MEMBER(x1, x2)
fork(x1, x2, x3) = fork(x1, x2, x3)

Lexicographic path order with status [19].
Precedence:

fork₃ > MEMBER₂

Status:

fork₃: multiset
MEMBER₂: [1,2]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

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

app(app(lt, app(s, x)), app(s, y)) → app(app(lt, x), y)
app(app(lt, 0), app(s, y)) → true
app(app(lt, y), 0) → false
app(app(eq, x), x) → true
app(app(eq, app(s, x)), 0) → false
app(app(eq, 0), app(s, x)) → false
app(app(member, w), null) → false
app(app(member, w), app(app(app(fork, x), y), z)) → app(app(app(if, app(app(lt, w), y)), app(app(member, w), x)), app(app(app(if, app(app(eq, w), y)), true), app(app(member, w), z)))

The set Q consists of the following terms:

app(app(lt, app(s, x0)), app(s, x1))
app(app(lt, 0), app(s, x0))
app(app(lt, x0), 0)
app(app(eq, x0), x0)
app(app(eq, app(s, x0)), 0)
app(app(eq, 0), app(s, x0))
app(app(member, x0), null)
app(app(member, x0), app(app(app(fork, x1), x2), x3))

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