Termination w.r.t. Q proof of TRS/higher-order/ToyamaRTA04Ex5Folding.trs

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

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

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

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

Q is empty.

The TRS is overlay and locally confluent. By [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(app(fold, f), x), nil) → x
app(app(app(fold, f), x), app(app(cons, y), z)) → app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(times, 0), y) → 0
app(app(times, app(s, x)), y) → app(app(plus, app(app(times, x), y)), y)
sum → app(app(fold, add), 0)
prod → app(app(fold, mul), app(s, 0))

The set Q consists of the following terms:

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

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 10 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(plus, app(s, x)), y) → APP(app(plus, x), y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

PLUS(s(x), y) → PLUS(x, y)

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

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

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

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [19].
Precedence:

s₁ > PLUS₁

Status:

s₁: multiset
PLUS₁: [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

                  ↳ QDP

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

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The 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(times, app(s, x)), y) → APP(app(times, x), y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

TIMES(s(x), y) → TIMES(x, y)

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

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

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

The following pairs can be oriented strictly and are deleted.

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

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

Lexicographic path order with status [19].
Precedence:

s₁ > TIMES₁

Status:

TIMES₁: [1]
s₁: multiset

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

The set Q consists of the following terms:

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

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

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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