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

(0) Obligation:

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

app(app(mapt, f), app(leaf, x)) → app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) → app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) → nil
app(app(maptlist, f), app(app(cons, x), xs)) → app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), 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(mapt, f), app(leaf, x)) → app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) → app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) → nil
app(app(maptlist, f), app(app(cons, x), xs)) → app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

The set Q consists of the following terms:

app(app(mapt, x0), app(leaf, x1))
app(app(mapt, x0), app(node, x1))
app(app(maptlist, x0), nil)
app(app(maptlist, 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(mapt, f), app(leaf, x)) → APP(leaf, app(f, x))
APP(app(mapt, f), app(leaf, x)) → APP(f, x)
APP(app(mapt, f), app(node, xs)) → APP(node, app(app(maptlist, f), xs))
APP(app(mapt, f), app(node, xs)) → APP(app(maptlist, f), xs)
APP(app(mapt, f), app(node, xs)) → APP(maptlist, f)
APP(app(maptlist, f), app(app(cons, x), xs)) → APP(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))
APP(app(maptlist, f), app(app(cons, x), xs)) → APP(cons, app(app(mapt, f), x))
APP(app(maptlist, f), app(app(cons, x), xs)) → APP(app(mapt, f), x)
APP(app(maptlist, f), app(app(cons, x), xs)) → APP(mapt, f)
APP(app(maptlist, f), app(app(cons, x), xs)) → APP(app(maptlist, f), xs)

The TRS R consists of the following rules:

app(app(mapt, f), app(leaf, x)) → app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) → app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) → nil
app(app(maptlist, f), app(app(cons, x), xs)) → app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

The set Q consists of the following terms:

app(app(mapt, x0), app(leaf, x1))
app(app(mapt, x0), app(node, x1))
app(app(maptlist, x0), nil)
app(app(maptlist, 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 6 less nodes.

(6) Obligation:

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

APP(app(mapt, f), app(node, xs)) → APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) → APP(app(mapt, f), x)
APP(app(mapt, f), app(leaf, x)) → APP(f, x)
APP(app(maptlist, f), app(app(cons, x), xs)) → APP(app(maptlist, f), xs)

The TRS R consists of the following rules:

app(app(mapt, f), app(leaf, x)) → app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) → app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) → nil
app(app(maptlist, f), app(app(cons, x), xs)) → app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

The set Q consists of the following terms:

app(app(mapt, x0), app(leaf, x1))
app(app(mapt, x0), app(node, x1))
app(app(maptlist, x0), nil)
app(app(maptlist, 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(mapt, f), app(node, xs)) → APP(app(maptlist, f), xs)
APP(app(maptlist, f), app(app(cons, x), xs)) → APP(app(mapt, f), x)
APP(app(mapt, f), app(leaf, x)) → APP(f, x)
APP(app(maptlist, f), app(app(cons, x), xs)) → APP(app(maptlist, f), xs)

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

Recursive Path Order [RPO].
Precedence:

APP₁ > maptlist
app₂ > maptlist
mapt > maptlist
node > maptlist
cons > maptlist
leaf > maptlist

The following usable rules [FROCOS05] were oriented: none

(8) Obligation:

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

app(app(mapt, f), app(leaf, x)) → app(leaf, app(f, x))
app(app(mapt, f), app(node, xs)) → app(node, app(app(maptlist, f), xs))
app(app(maptlist, f), nil) → nil
app(app(maptlist, f), app(app(cons, x), xs)) → app(app(cons, app(app(mapt, f), x)), app(app(maptlist, f), xs))

The set Q consists of the following terms:

app(app(mapt, x0), app(leaf, x1))
app(app(mapt, x0), app(node, x1))
app(app(maptlist, x0), nil)
app(app(maptlist, x0), app(app(cons, x1), x2))

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

(9) PisEmptyProof (EQUIVALENT transformation)

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

(10) TRUE