Termination w.r.t. Q proof of /tmp/tmph9HBS5/003.trs

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

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

app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(app(app(until, p), f), x) → app(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

The set Q consists of the following terms:

app(app(app(until, x0), x1), x2)
app(app(app(if, false), x0), x1)
app(app(app(if, true), x0), x1)

↳ QTRS

  ↳ DependencyPairsProof

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

app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(app(app(until, p), f), x) → app(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

The set Q consists of the following terms:

app(app(app(until, x0), x1), x2)
app(app(app(if, false), x0), x1)
app(app(app(if, true), x0), x1)

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

APP(app(app(until, p), f), x) → APP(app(app(until, p), f), app(f, x))
APP(app(app(until, p), f), x) → APP(f, x)
APP(app(app(until, p), f), x) → APP(p, x)
APP(app(app(until, p), f), x) → APP(app(if, app(p, x)), x)
APP(app(app(until, p), f), x) → APP(if, app(p, x))
APP(app(app(until, p), f), x) → APP(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

The TRS R consists of the following rules:

app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(app(app(until, p), f), x) → app(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

The set Q consists of the following terms:

app(app(app(until, x0), x1), x2)
app(app(app(if, false), x0), x1)
app(app(app(if, true), x0), x1)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

APP(app(app(until, p), f), x) → APP(app(app(until, p), f), app(f, x))
APP(app(app(until, p), f), x) → APP(f, x)
APP(app(app(until, p), f), x) → APP(p, x)
APP(app(app(until, p), f), x) → APP(app(if, app(p, x)), x)
APP(app(app(until, p), f), x) → APP(if, app(p, x))
APP(app(app(until, p), f), x) → APP(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

The TRS R consists of the following rules:

app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(app(app(until, p), f), x) → app(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

The set Q consists of the following terms:

app(app(app(until, x0), x1), x2)
app(app(app(if, false), x0), x1)
app(app(app(if, true), x0), x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

          ↳ QDPOrderProof

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

APP(app(app(until, p), f), x) → APP(app(app(until, p), f), app(f, x))
APP(app(app(until, p), f), x) → APP(f, x)
APP(app(app(until, p), f), x) → APP(p, x)

The TRS R consists of the following rules:

app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(app(app(until, p), f), x) → app(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

The set Q consists of the following terms:

app(app(app(until, x0), x1), x2)
app(app(app(if, false), x0), x1)
app(app(app(if, true), x0), x1)

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

The following pairs can be oriented strictly and are deleted.

APP(app(app(until, p), f), x) → APP(f, x)
APP(app(app(until, p), f), x) → APP(p, x)

The remaining pairs can at least be oriented weakly.

APP(app(app(until, p), f), x) → APP(app(app(until, p), f), app(f, x))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(APP(x₁, x₂)) = x₁
POL(app(x₁, x₂)) = 1 + x₁ + x₂
POL(false) = 0
POL(if) = 0
POL(true) = 0
POL(until) = 0

The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ MNOCProof

          ↳ QDPOrderProof

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

APP(app(app(until, p), f), x) → APP(app(app(until, p), f), app(f, x))

The TRS R consists of the following rules:

app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(app(app(until, p), f), x) → app(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

The set Q consists of the following terms:

app(app(app(until, x0), x1), x2)
app(app(app(if, false), x0), x1)
app(app(app(if, true), x0), x1)

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

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ MNOCProof

                ↳ QDP

                  ↳ NonTerminationProof

          ↳ QDPOrderProof

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

APP(app(app(until, p), f), x) → APP(app(app(until, p), f), app(f, x))

The TRS R consists of the following rules:

app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(app(app(until, p), f), x) → app(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

APP(app(app(until, p), f), x) → APP(app(app(until, p), f), app(f, x))

The TRS R consists of the following rules:

app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(app(app(until, p), f), x) → app(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

s = APP(app(app(until, p), f), x) evaluates to t =APP(app(app(until, p), f), app(f, x))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [x / app(f, x)]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP(app(app(until, p), f), x) to APP(app(app(until, p), f), app(f, x)).

We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

APP(app(app(until, p), f), x) → APP(f, x)
APP(app(app(until, p), f), x) → APP(p, x)

The remaining pairs can at least be oriented weakly.

APP(app(app(until, p), f), x) → APP(app(app(until, p), f), app(f, x))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( until ) =

/	1	\
\	0	/

M( if ) =

/	0	\
\	0	/

M( true ) =

/	0	\
\	0	/

M( false ) =

/	0	\
\	0	/

M( app(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	1	0	/

x₁

/	0	1	\
\	0	1	/

x₂

Tuple symbols:

M( APP(x₁, x₂) ) =

[

]

x₁

[

]

x₂

Matrix type:
We used a basic matrix type which is not further parametrizeable.

As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

          ↳ QDPOrderProof

            ↳ QDP

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

APP(app(app(until, p), f), x) → APP(app(app(until, p), f), app(f, x))

The TRS R consists of the following rules:

app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(app(app(until, p), f), x) → app(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, x)))

The set Q consists of the following terms:

app(app(app(until, x0), x1), x2)
app(app(app(if, false), x0), x1)
app(app(app(if, true), x0), x1)

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