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 UsableRulesProof (⇔)
    ↳7 QDP
    ↳8 UsableRulesReductionPairsProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 UsableRulesProof (⇔)
    ↳14 QDP
    ↳15 QDPSizeChangeProof (⇔)
    ↳16 TRUE

(0) Obligation:

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

app(app(append, nil), l) → l
app(app(append, app(app(cons, h), t)), l) → app(app(cons, h), app(app(append, t), l))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map, f), t))
app(app(append, app(app(append, l1), l2)), l3) → app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) → app(app(append, app(app(map, f), l1)), app(app(map, f), l2))

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(append, app(app(cons, h), t)), l) → APP(app(cons, h), app(app(append, t), l))
APP(app(append, app(app(cons, h), t)), l) → APP(app(append, t), l)
APP(app(append, app(app(cons, h), t)), l) → APP(append, t)
APP(app(map, f), app(app(cons, h), t)) → APP(app(cons, app(f, h)), app(app(map, f), t))
APP(app(map, f), app(app(cons, h), t)) → APP(cons, app(f, h))
APP(app(map, f), app(app(cons, h), t)) → APP(f, h)
APP(app(map, f), app(app(cons, h), t)) → APP(app(map, f), t)
APP(app(append, app(app(append, l1), l2)), l3) → APP(app(append, l1), app(app(append, l2), l3))
APP(app(append, app(app(append, l1), l2)), l3) → APP(app(append, l2), l3)
APP(app(append, app(app(append, l1), l2)), l3) → APP(append, l2)
APP(app(map, f), app(app(append, l1), l2)) → APP(app(append, app(app(map, f), l1)), app(app(map, f), l2))
APP(app(map, f), app(app(append, l1), l2)) → APP(append, app(app(map, f), l1))
APP(app(map, f), app(app(append, l1), l2)) → APP(app(map, f), l1)
APP(app(map, f), app(app(append, l1), l2)) → APP(app(map, f), l2)

The TRS R consists of the following rules:

app(app(append, nil), l) → l
app(app(append, app(app(cons, h), t)), l) → app(app(cons, h), app(app(append, t), l))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map, f), t))
app(app(append, app(app(append, l1), l2)), l3) → app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) → app(app(append, app(app(map, f), l1)), app(app(map, f), l2))

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 2 SCCs with 7 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

APP(app(append, app(app(append, l1), l2)), l3) → APP(app(append, l1), app(app(append, l2), l3))
APP(app(append, app(app(cons, h), t)), l) → APP(app(append, t), l)
APP(app(append, app(app(append, l1), l2)), l3) → APP(app(append, l2), l3)

The TRS R consists of the following rules:

app(app(append, nil), l) → l
app(app(append, app(app(cons, h), t)), l) → app(app(cons, h), app(app(append, t), l))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map, f), t))
app(app(append, app(app(append, l1), l2)), l3) → app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) → app(app(append, app(app(map, f), l1)), app(app(map, f), l2))

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

(6) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(7) Obligation:

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

APP(app(append, app(app(append, l1), l2)), l3) → APP(app(append, l1), app(app(append, l2), l3))
APP(app(append, app(app(cons, h), t)), l) → APP(app(append, t), l)
APP(app(append, app(app(append, l1), l2)), l3) → APP(app(append, l2), l3)

The TRS R consists of the following rules:

app(app(append, nil), l) → l
app(app(append, app(app(cons, h), t)), l) → app(app(cons, h), app(app(append, t), l))
app(app(append, app(app(append, l1), l2)), l3) → app(app(append, l1), app(app(append, l2), l3))

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

(8) UsableRulesReductionPairsProof (EQUIVALENT transformation)

First, we A-transformed [FROCOS05] the QDP-Problem. Then we obtain the following A-transformed DP problem.
The pairs P are:

append1(append(l1, l2), l3) → append1(l1, append(l2, l3))
append1(cons(h, t), l) → append1(t, l)
append1(append(l1, l2), l3) → append1(l2, l3)

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

append(nil, l) → l
append(cons(h, t), l) → cons(h, append(t, l))
append(append(l1, l2), l3) → append(l1, append(l2, l3))

Q is empty.

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

append1(append(l1, l2), l3) → append1(l1, append(l2, l3))
append1(cons(h, t), l) → append1(t, l)
append1(append(l1, l2), l3) → append1(l2, l3)
The following rules are removed from R:

append(nil, l) → l
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(append(x1, x2)) = 1 + x1 + x2   
POL(append1(x1, x2)) = 2·x1 + x2   
POL(cons(x1, x2)) = 2 + 2·x1 + x2   
POL(nil) = 0   

(9) Obligation:

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

append(cons(h, t), l) → cons(h, append(t, l))
append(append(l1, l2), l3) → append(l1, append(l2, l3))

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

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

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

APP(app(map, f), app(app(cons, h), t)) → APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) → APP(f, h)
APP(app(map, f), app(app(append, l1), l2)) → APP(app(map, f), l1)
APP(app(map, f), app(app(append, l1), l2)) → APP(app(map, f), l2)

The TRS R consists of the following rules:

app(app(append, nil), l) → l
app(app(append, app(app(cons, h), t)), l) → app(app(cons, h), app(app(append, t), l))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map, f), t))
app(app(append, app(app(append, l1), l2)), l3) → app(app(append, l1), app(app(append, l2), l3))
app(app(map, f), app(app(append, l1), l2)) → app(app(append, app(app(map, f), l1)), app(app(map, f), l2))

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

(13) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(14) Obligation:

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

APP(app(map, f), app(app(cons, h), t)) → APP(app(map, f), t)
APP(app(map, f), app(app(cons, h), t)) → APP(f, h)
APP(app(map, f), app(app(append, l1), l2)) → APP(app(map, f), l1)
APP(app(map, f), app(app(append, l1), l2)) → APP(app(map, f), l2)

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

(15) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • APP(app(map, f), app(app(cons, h), t)) → APP(app(map, f), t)
    The graph contains the following edges 1 >= 1, 2 > 2

  • APP(app(map, f), app(app(cons, h), t)) → APP(f, h)
    The graph contains the following edges 1 > 1, 2 > 2

  • APP(app(map, f), app(app(append, l1), l2)) → APP(app(map, f), l1)
    The graph contains the following edges 1 >= 1, 2 > 2

  • APP(app(map, f), app(app(append, l1), l2)) → APP(app(map, f), l2)
    The graph contains the following edges 1 >= 1, 2 > 2

(16) TRUE