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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
    ↳5 QDP
        ↳6 QDPOrderProof (⇔)
        ↳7 QDP
        ↳8 PisEmptyProof (⇔)
        ↳9 TRUE
    ↳10 QDP
        ↳11 QDPOrderProof (⇔)
        ↳12 QDP
        ↳13 PisEmptyProof (⇔)
        ↳14 TRUE
    ↳15 QDP
        ↳16 QDPOrderProof (⇔)
        ↳17 QDP
        ↳18 QDPOrderProof (⇔)
        ↳19 QDP
        ↳20 PisEmptyProof (⇔)
        ↳21 TRUE

(0) Obligation:

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

app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), 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(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
APP(app(le, app(s, x)), app(s, y)) → APP(le, x)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(if, app(app(le, x), y))
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(le, x), y)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(le, x)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(maxlist, y), ys)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(maxlist, y)
APP(height, app(app(node, x), xs)) → APP(s, app(app(maxlist, 0), app(app(map, height), xs)))
APP(height, app(app(node, x), xs)) → APP(app(maxlist, 0), app(app(map, height), xs))
APP(height, app(app(node, x), xs)) → APP(maxlist, 0)
APP(height, app(app(node, x), xs)) → APP(app(map, height), xs)
APP(height, app(app(node, x), xs)) → APP(map, height)

The TRS R consists of the following rules:

app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), 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 3 SCCs with 12 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)

The TRS R consists of the following rules:

app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x1, x2)  =  APP(x2)

Tags:
APP has tags [1,1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
le > [app2, s]

Status:
app2: multiset
le: multiset
s: multiset


The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(maxlist, y), ys)

The TRS R consists of the following rules:

app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(maxlist, y), ys)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x1, x2)  =  APP(x2)

Tags:
APP has tags [0,1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
app(x1, x2)  =  app(x2)
maxlist  =  maxlist
cons  =  cons

Recursive path order with status [RPO].
Quasi-Precedence:
maxlist > [app1, cons]

Status:
app1: multiset
maxlist: multiset
cons: multiset


The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(height, app(app(node, x), xs)) → APP(app(map, height), xs)

The TRS R consists of the following rules:

app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(height, app(app(node, x), xs)) → APP(app(map, height), xs)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x1, x2)  =  APP(x1, x2)

Tags:
APP has tags [1,0]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
[app2, node] > [map, height]

Status:
app2: multiset
map: multiset
cons: multiset
height: multiset
node: multiset


The following usable rules [FROCOS05] were oriented: none

(17) Obligation:

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

APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)

The TRS R consists of the following rules:

app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

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

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x1, x2)  =  APP(x2)

Tags:
APP has tags [0,1]

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Recursive path order with status [RPO].
Quasi-Precedence:
[app2, cons]

Status:
app2: multiset
map: multiset
cons: multiset


The following usable rules [FROCOS05] were oriented: none

(19) Obligation:

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

app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))

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

(20) PisEmptyProof (EQUIVALENT transformation)

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

(21) TRUE