Termination w.r.t. Q proof of /tmp/tmpju4Wwo/009.trs

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 Overlay + Local Confluence (⇔)

↳2 QTRS

↳3 DependencyPairsProof (⇔)

↳4 QDP

↳5 DependencyGraphProof (⇔)

↳6 QDP

↳7 QDPOrderProof (⇔)

↳8 QDP

↳9 DependencyGraphProof (⇔)

↳10 AND

  ↳11 QDP

    ↳12 QDPOrderProof (⇔)

    ↳13 QDP

    ↳14 PisEmptyProof (⇔)

    ↳15 TRUE

  ↳16 QDP

    ↳17 QDPOrderProof (⇔)

    ↳18 QDP

    ↳19 PisEmptyProof (⇔)

    ↳20 TRUE

(0) Obligation:

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.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(2) Obligation:

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))

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(6) Obligation:

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

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(p, x)
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.

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
APP(x1, x2) = APP(x1)
app(x1, x2) = app(x1, x2)
forall = forall
cons = cons
forsome = forsome

Recursive Path Order [RPO].
Precedence:

APP₁ > app₂
APP₁ > forall
APP₁ > forsome

The following usable rules [FROCOS05] were oriented: none

(8) Obligation:

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

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.

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(10) Complex Obligation (AND)

(11) Obligation:

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

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.

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04]. Here, we combined the reduction pair processor with the A-transformation [FROCOS05] which results in the following intermediate Q-DP Problem.
The a-transformed P is

forsome1(p, cons(x, xs)) → forsome1(p, xs)

The a-transformed usable rules are
none

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
forsome1(x1, x2) = forsome1(x2)
cons(x1, x2) = cons(x1, x2)

Recursive Path Order [RPO].
Precedence:

cons₂ > forsome1₁

The following usable rules [FROCOS05] were oriented: none

(13) Obligation:

Q DP problem:
P is empty.
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.

(14) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(15) TRUE

(16) Obligation:

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

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

(17) QDPOrderProof (EQUIVALENT transformation)

forall1(p, cons(x, xs)) → forall1(p, xs)

The a-transformed usable rules are
none

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
forall1(x1, x2) = forall1(x2)
cons(x1, x2) = cons(x1, x2)

Recursive Path Order [RPO].
Precedence:

cons₂ > forall1₁

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

Q DP problem:
P is empty.
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.

(19) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(0) Obligation:

(1) Overlay + Local Confluence (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) QDPOrderProof (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyGraphProof (EQUIVALENT transformation)

(10) Complex Obligation (AND)

(11) Obligation:

(12) QDPOrderProof (EQUIVALENT transformation)

(13) Obligation:

(14) PisEmptyProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) PisEmptyProof (EQUIVALENT transformation)

(20) TRUE