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, x), false) → false
app(app(and, false), y) → false
app(app(or, true), y) → true
app(app(or, x), 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, x), false) → false
app(app(and, false), y) → false
app(app(or, true), y) → true
app(app(or, x), 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, x0), false)
app(app(and, false), x0)
app(app(or, true), x0)
app(app(or, x0), 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, x), false) → false
app(app(and, false), y) → false
app(app(or, true), y) → true
app(app(or, x), 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, x0), false)
app(app(and, false), x0)
app(app(or, true), x0)
app(app(or, x0), 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, x), false) → false
app(app(and, false), y) → false
app(app(or, true), y) → true
app(app(or, x), 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, x0), false)
app(app(and, false), x0)
app(app(or, true), x0)
app(app(or, x0), 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)  =  x1
app(x1, x2)  =  app(x1, x2)
forall  =  forall
cons  =  cons
forsome  =  forsome

Recursive path order with status [RPO].
Precedence:
forall > app2
cons > app2
forsome > app2

Status:
cons: multiset
forall: multiset
app2: multiset
forsome: multiset

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, x), false) → false
app(app(and, false), y) → false
app(app(or, true), y) → true
app(app(or, x), 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, x0), false)
app(app(and, false), x0)
app(app(or, true), x0)
app(app(or, x0), 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, x), false) → false
app(app(and, false), y) → false
app(app(or, true), y) → true
app(app(or, x), 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, x0), false)
app(app(and, false), x0)
app(app(or, true), x0)
app(app(or, x0), 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: Recursive path order with status [RPO].
Precedence:
cons2 > forsome12

Status:
cons2: multiset
forsome12: [1,2]

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, x), false) → false
app(app(and, false), y) → false
app(app(or, true), y) → true
app(app(or, x), 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, x0), false)
app(app(and, false), x0)
app(app(or, true), x0)
app(app(or, x0), 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, x), false) → false
app(app(and, false), y) → false
app(app(or, true), y) → true
app(app(or, x), 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, x0), false)
app(app(and, false), x0)
app(app(or, true), x0)
app(app(or, x0), 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)

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

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: Recursive path order with status [RPO].
Precedence:
cons2 > forall12

Status:
cons2: multiset
forall12: [1,2]

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, x), false) → false
app(app(and, false), y) → false
app(app(or, true), y) → true
app(app(or, x), 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, x0), false)
app(app(and, false), x0)
app(app(or, true), x0)
app(app(or, x0), 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.

(20) TRUE