Termination w.r.t. Q proof of /tmp/tmpxNmpeU/009.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(and, true), true) → true
app(app(and, true), false) → false
app(app(and, false), true) → false
app(app(and, false), false) → false
app(app(or, true), true) → true
app(app(or, true), false) → true
app(app(or, false), true) → true
app(app(or, false), false) → false
app(app(forall, p), nil) → true
app(app(forall, p), app(app(cons, x), xs)) → app(app(and, app(p, x)), app(app(forall, p), xs))
app(app(forsome, p), nil) → false
app(app(forsome, p), app(app(cons, x), xs)) → app(app(or, app(p, x)), app(app(forsome, p), xs))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

app(app(and, true), true) → true
app(app(and, true), false) → false
app(app(and, false), true) → false
app(app(and, false), false) → false
app(app(or, true), true) → true
app(app(or, true), false) → true
app(app(or, false), true) → true
app(app(or, false), false) → false
app(app(forall, p), nil) → true
app(app(forall, p), app(app(cons, x), xs)) → app(app(and, app(p, x)), app(app(forall, p), xs))
app(app(forsome, p), nil) → false
app(app(forsome, p), app(app(cons, x), xs)) → app(app(or, app(p, x)), app(app(forsome, p), xs))

Q is empty.

The TRS is overlay and locally confluent. By [19] 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(and, true), true) → true
app(app(and, true), false) → false
app(app(and, false), true) → false
app(app(and, false), false) → false
app(app(or, true), true) → true
app(app(or, true), false) → true
app(app(or, false), true) → true
app(app(or, false), false) → false
app(app(forall, p), nil) → true
app(app(forall, p), app(app(cons, x), xs)) → app(app(and, app(p, x)), app(app(forall, p), xs))
app(app(forsome, p), nil) → false
app(app(forsome, p), app(app(cons, x), xs)) → app(app(or, app(p, x)), app(app(forsome, p), xs))

The set Q consists of the following terms:

app(app(and, true), true)
app(app(and, true), false)
app(app(and, false), true)
app(app(and, false), false)
app(app(or, true), true)
app(app(or, true), false)
app(app(or, false), true)
app(app(or, false), false)
app(app(forall, x0), nil)
app(app(forall, x0), app(app(cons, x1), x2))
app(app(forsome, x0), nil)
app(app(forsome, x0), app(app(cons, x1), x2))

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(forsome, p), app(app(cons, x), xs)) → APP(or, app(p, x))
APP(app(forsome, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(forall, p), app(app(cons, x), xs)) → APP(app(and, app(p, x)), app(app(forall, p), xs))
APP(app(forsome, p), app(app(cons, x), xs)) → APP(app(or, app(p, x)), app(app(forsome, p), xs))
APP(app(forall, p), app(app(cons, x), xs)) → APP(app(forall, p), xs)
APP(app(forall, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(forsome, p), app(app(cons, x), xs)) → APP(app(forsome, p), xs)
APP(app(forall, p), app(app(cons, x), xs)) → APP(and, app(p, x))

The TRS R consists of the following rules:

app(app(and, true), true) → true
app(app(and, true), false) → false
app(app(and, false), true) → false
app(app(and, false), false) → false
app(app(or, true), true) → true
app(app(or, true), false) → true
app(app(or, false), true) → true
app(app(or, false), false) → false
app(app(forall, p), nil) → true
app(app(forall, p), app(app(cons, x), xs)) → app(app(and, app(p, x)), app(app(forall, p), xs))
app(app(forsome, p), nil) → false
app(app(forsome, p), app(app(cons, x), xs)) → app(app(or, app(p, x)), app(app(forsome, p), xs))

The set Q consists of the following terms:

app(app(and, true), true)
app(app(and, true), false)
app(app(and, false), true)
app(app(and, false), false)
app(app(or, true), true)
app(app(or, true), false)
app(app(or, false), true)
app(app(or, false), false)
app(app(forall, x0), nil)
app(app(forall, x0), app(app(cons, x1), x2))
app(app(forsome, x0), nil)
app(app(forsome, x0), app(app(cons, x1), x2))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

APP(app(forsome, p), app(app(cons, x), xs)) → APP(or, app(p, x))
APP(app(forsome, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(forall, p), app(app(cons, x), xs)) → APP(app(and, app(p, x)), app(app(forall, p), xs))
APP(app(forsome, p), app(app(cons, x), xs)) → APP(app(or, app(p, x)), app(app(forsome, p), xs))
APP(app(forall, p), app(app(cons, x), xs)) → APP(app(forall, p), xs)
APP(app(forall, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(forsome, p), app(app(cons, x), xs)) → APP(app(forsome, p), xs)
APP(app(forall, p), app(app(cons, x), xs)) → APP(and, app(p, x))

The TRS R consists of the following rules:

app(app(and, true), true) → true
app(app(and, true), false) → false
app(app(and, false), true) → false
app(app(and, false), false) → false
app(app(or, true), true) → true
app(app(or, true), false) → true
app(app(or, false), true) → true
app(app(or, false), false) → false
app(app(forall, p), nil) → true
app(app(forall, p), app(app(cons, x), xs)) → app(app(and, app(p, x)), app(app(forall, p), xs))
app(app(forsome, p), nil) → false
app(app(forsome, p), app(app(cons, x), xs)) → app(app(or, app(p, x)), app(app(forsome, p), xs))

The set Q consists of the following terms:

app(app(and, true), true)
app(app(and, true), false)
app(app(and, false), true)
app(app(and, false), false)
app(app(or, true), true)
app(app(or, true), false)
app(app(or, false), true)
app(app(or, false), false)
app(app(forall, x0), nil)
app(app(forall, x0), app(app(cons, x1), x2))
app(app(forsome, x0), nil)
app(app(forsome, x0), app(app(cons, x1), x2))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ UsableRulesProof

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

APP(app(forsome, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(forall, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(forall, p), app(app(cons, x), xs)) → APP(app(forall, p), xs)
APP(app(forsome, p), app(app(cons, x), xs)) → APP(app(forsome, p), xs)

The TRS R consists of the following rules:

app(app(and, true), true) → true
app(app(and, true), false) → false
app(app(and, false), true) → false
app(app(and, false), false) → false
app(app(or, true), true) → true
app(app(or, true), false) → true
app(app(or, false), true) → true
app(app(or, false), false) → false
app(app(forall, p), nil) → true
app(app(forall, p), app(app(cons, x), xs)) → app(app(and, app(p, x)), app(app(forall, p), xs))
app(app(forsome, p), nil) → false
app(app(forsome, p), app(app(cons, x), xs)) → app(app(or, app(p, x)), app(app(forsome, p), xs))

The set Q consists of the following terms:

app(app(and, true), true)
app(app(and, true), false)
app(app(and, false), true)
app(app(and, false), false)
app(app(or, true), true)
app(app(or, true), false)
app(app(or, false), true)
app(app(or, false), false)
app(app(forall, x0), nil)
app(app(forall, x0), app(app(cons, x1), x2))
app(app(forsome, x0), nil)
app(app(forsome, x0), app(app(cons, x1), x2))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ UsableRulesProof

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

APP(app(forsome, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(forall, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(forall, p), app(app(cons, x), xs)) → APP(app(forall, p), xs)
APP(app(forsome, p), app(app(cons, x), xs)) → APP(app(forsome, p), xs)

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

app(app(and, true), true)
app(app(and, true), false)
app(app(and, false), true)
app(app(and, false), false)
app(app(or, true), true)
app(app(or, true), false)
app(app(or, false), true)
app(app(or, false), false)
app(app(forall, x0), nil)
app(app(forall, x0), app(app(cons, x1), x2))
app(app(forsome, x0), nil)
app(app(forsome, x0), app(app(cons, x1), x2))

We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

APP(app(forsome, p), app(app(cons, x), xs)) → APP(p, x)
The graph contains the following edges 1 > 1, 2 > 2
APP(app(forsome, p), app(app(cons, x), xs)) → APP(app(forsome, p), xs)
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(forall, p), app(app(cons, x), xs)) → APP(p, x)
The graph contains the following edges 1 > 1, 2 > 2
APP(app(forall, p), app(app(cons, x), xs)) → APP(app(forall, p), xs)
The graph contains the following edges 1 >= 1, 2 > 2