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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 QDPOrderProof (⇔)
↳6 QDP
↳7 PisEmptyProof (⇔)
↳8 TRUE

(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, true), x), y) → y
app(app(takeWhile, p), nil) → nil
app(app(takeWhile, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(takeWhile, p), xs))), nil)
app(app(dropWhile, p), nil) → nil
app(app(dropWhile, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(dropWhile, p), xs)), app(app(cons, x), xs))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

APP(app(takeWhile, p), app(app(cons, x), xs)) → APP(app(app(if, app(p, x)), app(app(cons, x), app(app(takeWhile, p), xs))), nil)
APP(app(takeWhile, p), app(app(cons, x), xs)) → APP(app(if, app(p, x)), app(app(cons, x), app(app(takeWhile, p), xs)))
APP(app(takeWhile, p), app(app(cons, x), xs)) → APP(if, app(p, x))
APP(app(takeWhile, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(takeWhile, p), app(app(cons, x), xs)) → APP(app(cons, x), app(app(takeWhile, p), xs))
APP(app(takeWhile, p), app(app(cons, x), xs)) → APP(app(takeWhile, p), xs)
APP(app(dropWhile, p), app(app(cons, x), xs)) → APP(app(app(if, app(p, x)), app(app(dropWhile, p), xs)), app(app(cons, x), xs))
APP(app(dropWhile, p), app(app(cons, x), xs)) → APP(app(if, app(p, x)), app(app(dropWhile, p), xs))
APP(app(dropWhile, p), app(app(cons, x), xs)) → APP(if, app(p, x))
APP(app(dropWhile, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(dropWhile, p), app(app(cons, x), xs)) → APP(app(dropWhile, p), xs)

The TRS R consists of the following rules:

app(app(app(if, true), x), y) → x
app(app(app(if, true), x), y) → y
app(app(takeWhile, p), nil) → nil
app(app(takeWhile, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(takeWhile, p), xs))), nil)
app(app(dropWhile, p), nil) → nil
app(app(dropWhile, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(dropWhile, p), xs)), app(app(cons, x), xs))

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.

(4) Obligation:

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

APP(app(takeWhile, p), app(app(cons, x), xs)) → APP(app(takeWhile, p), xs)
APP(app(takeWhile, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(dropWhile, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(dropWhile, p), app(app(cons, x), xs)) → APP(app(dropWhile, p), xs)

The TRS R consists of the following rules:

app(app(app(if, true), x), y) → x
app(app(app(if, true), x), y) → y
app(app(takeWhile, p), nil) → nil
app(app(takeWhile, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(takeWhile, p), xs))), nil)
app(app(dropWhile, p), nil) → nil
app(app(dropWhile, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(dropWhile, p), xs)), app(app(cons, x), xs))

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(app(takeWhile, p), app(app(cons, x), xs)) → APP(app(takeWhile, p), xs)
APP(app(takeWhile, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(dropWhile, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(dropWhile, p), app(app(cons, x), xs)) → APP(app(dropWhile, p), xs)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
[APP2, dropWhile] > takeWhile > [app2, cons]


The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

app(app(app(if, true), x), y) → x
app(app(app(if, true), x), y) → y
app(app(takeWhile, p), nil) → nil
app(app(takeWhile, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(takeWhile, p), xs))), nil)
app(app(dropWhile, p), nil) → nil
app(app(dropWhile, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(dropWhile, p), xs)), app(app(cons, x), xs))

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

(7) PisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE