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 QDPOrderProof (⇔)
↳8 QDP
↳9 DependencyGraphProof (⇔)
↳10 QDP
↳11 QDPOrderProof (⇔)
↳12 QDP
↳13 PisEmptyProof (⇔)
↳14 TRUE

(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) 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(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))

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 6 less nodes.

(4) 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))

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(mapt, f), app(leaf, x)) → APP(f, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(APP(x1, x2)) = x1   
POL(app(x1, x2)) = 1 + x2   
POL(cons) = 0   
POL(leaf) = 0   
POL(mapt) = 0   
POL(maptlist) = 0   
POL(node) = 0   

The following usable rules [FROCOS05] were oriented: none

(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(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))

Q is empty.
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)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(APP(x1, x2)) = x2   
POL(app(x1, x2)) = x1 + x2   
POL(cons) = 0   
POL(mapt) = 0   
POL(maptlist) = 0   
POL(node) = 1   

The following usable rules [FROCOS05] were oriented: none

(8) Obligation:

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

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

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

(9) DependencyGraphProof (EQUIVALENT transformation)

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

(10) Obligation:

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

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))

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(maptlist, f), app(app(cons, x), xs)) → APP(app(maptlist, f), xs)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(APP(x1, x2)) = x2   
POL(app(x1, x2)) = 1 + x1 + x2   
POL(cons) = 0   
POL(maptlist) = 0   

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(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.
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