Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

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)))

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(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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(app(app(until, x0), 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(app(until, p), f), x) → APP(app(app(if, app(p, x)), x), app(app(app(until, p), f), app(f, 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(p, x)
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)

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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(app(app(until, x0), 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 3 less nodes.

(6) Obligation:

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(p, x)
APP(app(app(until, p), f), x) → 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(app(app(until, x0), 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(app(until, p), f), x) → APP(p, x)
APP(app(app(until, p), f), x) → APP(f, 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)
until  =  until
if  =  if
false  =  false
true  =  true

Lexicographic path order with status [LPO].
Precedence:

APP₁ > until
app₂ > until
app₂ > if

Status:

APP₁: [1]
until: []
if: []
true: []
false: []
app₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(8) Obligation:

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

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