Termination w.r.t. Q proof of TRS/currying/D3331.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(:, app(app(:, x), y)), z) → app(app(:, x), app(app(:, y), z))
app(app(:, app(app(+, x), y)), z) → app(app(+, app(app(:, x), z)), app(app(:, y), z))
app(app(:, z), app(app(+, x), app(f, y))) → app(app(:, app(app(g, z), y)), app(app(+, x), a))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

app(app(:, app(app(:, x), y)), z) → app(app(:, x), app(app(:, y), z))
app(app(:, app(app(+, x), y)), z) → app(app(+, app(app(:, x), z)), app(app(:, y), z))
app(app(:, z), app(app(+, x), app(f, y))) → app(app(:, app(app(g, z), y)), app(app(+, x), a))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

Q is empty.

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(:, z), app(app(+, x), app(f, y))) → APP(:, app(app(g, z), y))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(:, z), app(app(+, x), app(f, y))) → APP(app(:, app(app(g, z), y)), app(app(+, x), a))
APP(app(:, app(app(:, x), y)), z) → APP(app(:, y), z)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(:, app(app(+, x), y)), z) → APP(:, y)
APP(app(:, z), app(app(+, x), app(f, y))) → APP(app(+, x), a)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)
APP(app(:, z), app(app(+, x), app(f, y))) → APP(app(g, z), y)
APP(app(:, app(app(+, x), y)), z) → APP(app(:, y), z)
APP(app(:, app(app(+, x), y)), z) → APP(app(+, app(app(:, x), z)), app(app(:, y), z))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(:, app(app(+, x), y)), z) → APP(+, app(app(:, x), z))
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(app(:, z), app(app(+, x), app(f, y))) → APP(g, z)
APP(app(:, app(app(:, x), y)), z) → APP(app(:, x), app(app(:, y), z))
APP(app(:, app(app(+, x), y)), z) → APP(app(:, x), z)
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(:, app(app(:, x), y)), z) → APP(:, y)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(:, app(app(+, x), y)), z) → APP(:, x)

The TRS R consists of the following rules:

app(app(:, app(app(:, x), y)), z) → app(app(:, x), app(app(:, y), z))
app(app(:, app(app(+, x), y)), z) → app(app(+, app(app(:, x), z)), app(app(:, y), z))
app(app(:, z), app(app(+, x), app(f, y))) → app(app(:, app(app(g, z), y)), app(app(+, x), a))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

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

APP(app(:, z), app(app(+, x), app(f, y))) → APP(:, app(app(g, z), y))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(:, z), app(app(+, x), app(f, y))) → APP(app(:, app(app(g, z), y)), app(app(+, x), a))
APP(app(:, app(app(:, x), y)), z) → APP(app(:, y), z)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(:, app(app(+, x), y)), z) → APP(:, y)
APP(app(:, z), app(app(+, x), app(f, y))) → APP(app(+, x), a)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)
APP(app(:, z), app(app(+, x), app(f, y))) → APP(app(g, z), y)
APP(app(:, app(app(+, x), y)), z) → APP(app(:, y), z)
APP(app(:, app(app(+, x), y)), z) → APP(app(+, app(app(:, x), z)), app(app(:, y), z))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(:, app(app(+, x), y)), z) → APP(+, app(app(:, x), z))
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(app(:, z), app(app(+, x), app(f, y))) → APP(g, z)
APP(app(:, app(app(:, x), y)), z) → APP(app(:, x), app(app(:, y), z))
APP(app(:, app(app(+, x), y)), z) → APP(app(:, x), z)
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(:, app(app(:, x), y)), z) → APP(:, y)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(:, app(app(+, x), y)), z) → APP(:, x)

The TRS R consists of the following rules:

app(app(:, app(app(:, x), y)), z) → app(app(:, x), app(app(:, y), z))
app(app(:, app(app(+, x), y)), z) → app(app(+, app(app(:, x), z)), app(app(:, y), z))
app(app(:, z), app(app(+, x), app(f, y))) → app(app(:, app(app(g, z), y)), app(app(+, x), a))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

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

APP(app(:, z), app(app(+, x), app(f, y))) → APP(:, app(app(g, z), y))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(:, z), app(app(+, x), app(f, y))) → APP(app(:, app(app(g, z), y)), app(app(+, x), a))
APP(app(:, app(app(:, x), y)), z) → APP(app(:, y), z)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(:, app(app(+, x), y)), z) → APP(:, y)
APP(app(:, z), app(app(+, x), app(f, y))) → APP(app(+, x), a)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)
APP(app(:, z), app(app(+, x), app(f, y))) → APP(app(g, z), y)
APP(app(filter, 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(:, app(app(+, x), y)), z) → APP(app(+, app(app(:, x), z)), app(app(:, y), z))
APP(app(:, app(app(+, x), y)), z) → APP(app(:, y), z)
APP(app(:, app(app(+, x), y)), z) → APP(+, app(app(:, x), z))
APP(app(:, z), app(app(+, x), app(f, y))) → APP(g, z)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(app(:, app(app(:, x), y)), z) → APP(app(:, x), app(app(:, y), z))
APP(app(:, app(app(+, x), y)), z) → APP(app(:, x), z)
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(:, app(app(:, x), y)), z) → APP(:, y)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(:, app(app(+, x), y)), z) → APP(:, x)

The TRS R consists of the following rules:

app(app(:, app(app(:, x), y)), z) → app(app(:, x), app(app(:, y), z))
app(app(:, app(app(+, x), y)), z) → app(app(+, app(app(:, x), z)), app(app(:, y), z))
app(app(:, z), app(app(+, x), app(f, y))) → app(app(:, app(app(g, z), y)), app(app(+, x), a))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 20 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

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

APP(app(:, app(app(:, x), y)), z) → APP(app(:, x), app(app(:, y), z))
APP(app(:, app(app(+, x), y)), z) → APP(app(:, x), z)
APP(app(:, app(app(:, x), y)), z) → APP(app(:, y), z)
APP(app(:, app(app(+, x), y)), z) → APP(app(:, y), z)

The TRS R consists of the following rules:

app(app(:, app(app(:, x), y)), z) → app(app(:, x), app(app(:, y), z))
app(app(:, app(app(+, x), y)), z) → app(app(+, app(app(:, x), z)), app(app(:, y), z))
app(app(:, z), app(app(+, x), app(f, y))) → app(app(:, app(app(g, z), y)), app(app(+, x), a))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

Q is empty.
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(app(:, app(app(:, x), y)), z) → APP(app(:, x), app(app(:, y), z))
APP(app(:, app(app(+, x), y)), z) → APP(app(:, x), z)
APP(app(:, app(app(:, x), y)), z) → APP(app(:, y), z)
APP(app(:, app(app(+, x), y)), z) → APP(app(:, y), z)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
APP(x1, x2) = APP(x1)
app(x1, x2) = app(x1, x2)
: = :
+ = +
g = g
a = a

Recursive path order with status [2].
Quasi-Precedence:

+ > app₂ > [APP₁, :, a] > g

Status:

APP₁: multiset
a: multiset
:: multiset
app₂: multiset
+: multiset
g: multiset

The following usable rules [14] were oriented: none

↳ 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(:, app(app(:, x), y)), z) → app(app(:, x), app(app(:, y), z))
app(app(:, app(app(+, x), y)), z) → app(app(+, app(app(:, x), z)), app(app(:, y), z))
app(app(:, z), app(app(+, x), app(f, y))) → app(app(:, app(app(g, z), y)), app(app(+, x), a))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

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

APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(filter, 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(:, app(app(:, x), y)), z) → app(app(:, x), app(app(:, y), z))
app(app(:, app(app(+, x), y)), z) → app(app(+, app(app(:, x), z)), app(app(:, y), z))
app(app(:, z), app(app(+, x), app(f, y))) → app(app(:, app(app(g, z), y)), app(app(+, x), a))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

Q is empty.
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(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, 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.

APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)

Used ordering: Combined order from the following AFS and order.
APP(x1, x2) = x2
app(x1, x2) = app(x1, x2)
filter2 = filter2
true = true
filter = filter
map = map
cons = cons
false = false

Recursive path order with status [2].
Quasi-Precedence:

map > [app₂, filter2, cons, false]

Status:

filter: multiset
true: multiset
false: multiset
map: multiset
app₂: multiset
filter2: multiset
cons: multiset

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

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

APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)

The TRS R consists of the following rules:

app(app(:, app(app(:, x), y)), z) → app(app(:, x), app(app(:, y), z))
app(app(:, app(app(+, x), y)), z) → app(app(+, app(app(:, x), z)), app(app(:, y), z))
app(app(:, z), app(app(+, x), app(f, y))) → app(app(:, app(app(g, z), y)), app(app(+, x), a))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 2 less nodes.