Termination w.r.t. Q proof of /tmp/tmpudFIvv/Ex9Maps.trs

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

0 QTRS

↳1 Overlay + Local Confluence (⇔)

↳2 QTRS

↳3 DependencyPairsProof (⇔)

↳4 QDP

↳5 DependencyGraphProof (⇔)

↳6 QDP

↳7 ForwardInstantiation (⇔)

↳8 QDP

↳9 ForwardInstantiation (⇔)

↳10 QDP

↳11 ForwardInstantiation (⇔)

↳12 QDP

↳13 ForwardInstantiation (⇔)

↳14 QDP

↳15 ForwardInstantiation (⇔)

↳16 QDP

↳17 ForwardInstantiation (⇔)

↳18 QDP

↳19 UsableRulesProof (⇔)

↳20 QDP

↳21 ForwardInstantiation (⇔)

↳22 QDP

↳23 ForwardInstantiation (⇔)

↳24 QDP

↳25 MRRProof (⇔)

↳26 QDP

↳27 DependencyGraphProof (⇔)

↳28 AND

  ↳29 QDP

    ↳30 ATransformationProof (⇔)

    ↳31 QDP

    ↳32 QReductionProof (⇔)

    ↳33 QDP

    ↳34 QDPSizeChangeProof (⇔)

    ↳35 TRUE

  ↳36 QDP

    ↳37 MRRProof (⇔)

    ↳38 QDP

    ↳39 DependencyGraphProof (⇔)

    ↳40 AND

      ↳41 QDP

        ↳42 QDPSizeChangeProof (⇔)

        ↳43 TRUE

      ↳44 QDP

        ↳45 MRRProof (⇔)

        ↳46 QDP

        ↳47 QDPSizeChangeProof (⇔)

        ↳48 TRUE

(0) Obligation:

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

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

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(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

(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(map_1, f), app(app(cons, h), t)) → APP(app(cons, app(f, h)), app(app(map_1, f), t))
APP(app(map_1, f), app(app(cons, h), t)) → APP(cons, app(f, h))
APP(app(map_1, f), app(app(cons, h), t)) → APP(f, h)
APP(app(map_1, f), app(app(cons, h), t)) → APP(app(map_1, f), t)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(cons, app(app(f, h), c))
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(f, h), c)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(f, h)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(app(map_2, f), c), t)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(cons, app(app(app(f, g), h), c))
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(f, g)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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 7 less nodes.

(6) Obligation:

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

APP(app(map_1, f), app(app(cons, h), t)) → APP(app(map_1, f), t)
APP(app(map_1, f), app(app(cons, h), t)) → APP(f, h)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(f, h), c)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(f, h)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(app(map_2, f), c), t)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(7) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule APP(app(map_1, f), app(app(cons, h), t)) → APP(app(map_1, f), t) we obtained the following new rules [LPAR04]:

APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))

(8) Obligation:

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

APP(app(map_1, f), app(app(cons, h), t)) → APP(f, h)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(f, h), c)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(f, h)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(app(map_2, f), c), t)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(9) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule APP(app(map_1, f), app(app(cons, h), t)) → APP(f, h) we obtained the following new rules [LPAR04]:

APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), y_2))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))

(10) Obligation:

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

APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(f, h), c)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(f, h)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(app(map_2, f), c), t)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))
APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), y_2))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(11) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(f, h), c) we obtained the following new rules [LPAR04]:

APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, y_0), x2), x1)
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, x2), x3)) → APP(app(map_1, x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_0), y_1)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_0), g), y_1)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_0)), x3)) → APP(app(map_1, app(map_1, y_0)), x1)

(12) Obligation:

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

APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(f, h)
APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(app(map_2, f), c), t)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))
APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), y_2))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, y_0), x2), x1)
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, x2), x3)) → APP(app(map_1, x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_0), y_1)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_0), g), y_1)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_0)), x3)) → APP(app(map_1, app(map_1, y_0)), x1)

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(13) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(f, h) we obtained the following new rules [LPAR04]:

APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), y_2)), y_3)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, y_1), y_2))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(map_1, y_1)), y_2))

(14) Obligation:

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

APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(app(map_2, f), c), t)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))
APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), y_2))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, y_0), x2), x1)
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, x2), x3)) → APP(app(map_1, x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_0), y_1)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_0), g), y_1)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_0)), x3)) → APP(app(map_1, app(map_1, y_0)), x1)
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), y_2)), y_3)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, y_1), y_2))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(map_1, y_1)), y_2))

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(15) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule APP(app(app(map_2, f), c), app(app(cons, h), t)) → APP(app(app(map_2, f), c), t) we obtained the following new rules [LPAR04]:

APP(app(app(map_2, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, x0), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(map_2, y_1), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(map_1, y_1)), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))

(16) Obligation:

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

APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))
APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), y_2))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, y_0), x2), x1)
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, x2), x3)) → APP(app(map_1, x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_0), y_1)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_0), g), y_1)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_0)), x3)) → APP(app(map_1, app(map_1, y_0)), x1)
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), y_2)), y_3)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, y_1), y_2))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, x0), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(map_2, y_1), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(map_1, y_1)), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(17) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(f, g), h), c) we obtained the following new rules [LPAR04]:

APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, g), x2), x1)

(18) Obligation:

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

APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))
APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), y_2))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, y_0), x2), x1)
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, x2), x3)) → APP(app(map_1, x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_0), y_1)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_0), g), y_1)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_0)), x3)) → APP(app(map_1, app(map_1, y_0)), x1)
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), y_2)), y_3)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, y_1), y_2))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, x0), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(map_2, y_1), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(map_1, y_1)), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, g), x2), x1)

The TRS R consists of the following rules:

app(app(map_1, f), app(app(cons, h), t)) → app(app(cons, app(f, h)), app(app(map_1, f), t))
app(app(app(map_2, f), c), app(app(cons, h), t)) → app(app(cons, app(app(f, h), c)), app(app(app(map_2, f), c), t))
app(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → app(app(cons, app(app(app(f, g), h), c)), app(app(app(app(map_3, f), g), c), t))

The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(19) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(20) Obligation:

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

APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h)
APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))
APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), y_2))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, y_0), x2), x1)
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, x2), x3)) → APP(app(map_1, x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_0), y_1)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_0), g), y_1)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_0)), x3)) → APP(app(map_1, app(map_1, y_0)), x1)
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), y_2)), y_3)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, y_1), y_2))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, x0), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(map_2, y_1), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(map_1, y_1)), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, g), x2), x1)

R is empty.
The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(21) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(f, g), h) we obtained the following new rules [LPAR04]:

APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, g), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, y_0), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5)))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, app(map_3, y_0)), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(app(map_3, map_2), g), g), app(app(cons, y_1), y_2))

(22) Obligation:

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

APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t)
APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))
APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), y_2))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, y_0), x2), x1)
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, x2), x3)) → APP(app(map_1, x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_0), y_1)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_0), g), y_1)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_0)), x3)) → APP(app(map_1, app(map_1, y_0)), x1)
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), y_2)), y_3)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, y_1), y_2))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, x0), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(map_2, y_1), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(map_1, y_1)), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, g), x2), x1)
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, g), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, y_0), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5)))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, app(map_3, y_0)), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(app(map_3, map_2), g), g), app(app(cons, y_1), y_2))

R is empty.
The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(23) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule APP(app(app(app(map_3, f), g), c), app(app(cons, h), t)) → APP(app(app(app(map_3, f), g), c), t) we obtained the following new rules [LPAR04]:

APP(app(app(app(map_3, x0), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, x0), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(app(map_3, map_2), g), x1), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))

(24) Obligation:

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

APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))
APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), y_2))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, y_0), x2), x1)
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, x2), x3)) → APP(app(map_1, x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_0), y_1)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_0), g), y_1)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_0)), x3)) → APP(app(map_1, app(map_1, y_0)), x1)
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), y_2)), y_3)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, y_1), y_2))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, x0), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(map_2, y_1), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(map_1, y_1)), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, g), x2), x1)
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, g), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, y_0), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5)))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, app(map_3, y_0)), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(app(map_3, map_2), g), g), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, x0), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, x0), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(app(map_3, map_2), g), x1), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))

R is empty.
The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(25) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), x2)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), y_2))
APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x2)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))

Used ordering: Polynomial interpretation [POLO]:

POL(APP(x₁, x₂)) = x₁ + x₂
POL(app(x₁, x₂)) = x₁ + x₂
POL(cons) = 0
POL(g) = 0
POL(map_1) = 1
POL(map_2) = 0
POL(map_3) = 0

(26) Obligation:

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

APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, y_0), x2), x1)
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, x2), x3)) → APP(app(map_1, x2), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_0), y_1)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_0), g), y_1)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), x1)
APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_0)), x3)) → APP(app(map_1, app(map_1, y_0)), x1)
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, y_0), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(map_2, y_0), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(map_1, app(app(app(map_3, y_0), g), y_1)), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), y_2)), y_3)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4)), x3)) → APP(app(map_1, app(map_1, y_0)), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, y_1), y_2))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, x0), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(map_2, y_1), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(map_1, y_1)), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, g), x2), x1)
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(map_1, g), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, y_0), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5)))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, app(map_3, y_0)), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(app(map_3, map_2), g), g), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, x0), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, x0), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(app(map_3, map_2), g), x1), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))

R is empty.
The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 10 less nodes.

(28) Complex Obligation (AND)

(29) Obligation:

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

APP(app(map_1, x0), app(app(cons, x1), app(app(cons, y_1), y_2))) → APP(app(map_1, x0), app(app(cons, y_1), y_2))

R is empty.
The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(30) ATransformationProof (EQUIVALENT transformation)

We have applied the A-Transformation [FROCOS05] to get from an applicative problem to a standard problem.

(31) Obligation:

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

map_11(x0, cons(x1, cons(y_1, y_2))) → map_11(x0, cons(y_1, y_2))

R is empty.
The set Q consists of the following terms:

map_1(x0, cons(x1, x2))
map_2(x0, x1, cons(x2, x3))
map_3(x0, g, x1, cons(x2, x3))

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

(32) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

map_1(x0, cons(x1, x2))
map_2(x0, x1, cons(x2, x3))
map_3(x0, g, x1, cons(x2, x3))

(33) Obligation:

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

map_11(x0, cons(x1, cons(y_1, y_2))) → map_11(x0, cons(y_1, y_2))

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

(34) 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:

map_11(x0, cons(x1, cons(y_1, y_2))) → map_11(x0, cons(y_1, y_2))
The graph contains the following edges 1 >= 1, 2 > 2

(35) TRUE

(36) Obligation:

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

APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, y_0), x2), x1)
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, g), x2), x1)
APP(app(app(map_2, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, x0), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(map_2, y_1), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(map_1, y_1)), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, y_1), y_2))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, y_0), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5)))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, app(map_3, y_0)), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, x0), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, x0), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(app(map_3, map_2), g), g), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(app(map_3, map_2), g), x1), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))

R is empty.
The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(37) MRRProof (EQUIVALENT transformation)

APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, y_0), x2), x1)
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, y_0), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), y_1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, y_1), y_2))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), y_0), app(app(cons, app(map_1, y_1)), y_2))

Used ordering: Polynomial interpretation [POLO]:

POL(APP(x₁, x₂)) = x₁ + x₂
POL(app(x₁, x₂)) = x₁ + x₂
POL(cons) = 0
POL(g) = 0
POL(map_1) = 0
POL(map_2) = 1
POL(map_3) = 0

(38) Obligation:

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

APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), x3)) → APP(app(app(map_2, g), x2), x1)
APP(app(app(map_2, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, x0), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(map_2, y_1), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(map_1, y_1)), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, y_0), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(map_2, y_0)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), x3)) → APP(app(app(map_2, app(app(map_3, y_0), g)), g), app(app(cons, y_2), app(app(cons, y_3), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, y_2), y_3)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), x3)) → APP(app(app(map_2, map_1), g), app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), x3)) → APP(app(app(map_2, app(map_1, y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), g), app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), x3)) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), g), app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6)))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), x3)) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), g), app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5)))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, app(map_3, y_0)), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, x0), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, x0), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(app(map_3, map_2), g), g), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(app(map_3, map_2), g), x1), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))

R is empty.
The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(39) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 40 less nodes.

(40) Complex Obligation (AND)

(41) Obligation:

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

APP(app(app(map_2, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, x0), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, y_2), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, y_1), y_2))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(map_2, y_1), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(map_1, y_1)), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_1)), y_2))
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))

R is empty.
The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(42) 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(app(map_2, x0), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, x0), x1), app(app(cons, y_2), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(map_2, y_0)), x1), app(app(cons, y_2), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(map_2, app(app(map_3, y_0), g)), x1), app(app(cons, y_2), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, y_0), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(map_2, y_0)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), y_4)), y_5))) → APP(app(app(map_2, app(app(app(map_3, y_0), g), y_1)), x1), app(app(cons, app(app(cons, y_3), y_4)), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(map_2, app(map_1, y_0)), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(map_2, y_0), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1))), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, y_1), y_2))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(map_2, y_1), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(map_2, y_1), y_2)), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, map_1), x1), app(app(cons, x2), app(app(cons, app(map_1, y_1)), y_2))) → APP(app(app(map_2, map_1), x1), app(app(cons, app(map_1, y_1)), y_2))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))) → APP(app(app(map_2, app(app(map_2, map_1), y_0)), x1), app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(map_2, app(map_1, app(map_1, y_0))), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
The graph contains the following edges 1 >= 1, 2 > 2

(43) TRUE

(44) Obligation:

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

APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, app(map_3, y_0)), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, x0), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, x0), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(app(map_3, map_2), g), g), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(app(map_3, map_2), g), x1), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))

R is empty.
The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(45) MRRProof (EQUIVALENT transformation)

APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), x3)) → APP(app(app(app(map_3, y_0), g), x2), x1)
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, y_0), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), x3)) → APP(app(app(app(map_3, app(map_3, y_0)), g), g), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), x3)) → APP(app(app(app(map_3, map_2), g), g), app(app(cons, y_1), y_2))

Used ordering: Polynomial interpretation [POLO]:

POL(APP(x₁, x₂)) = x₁ + x₂
POL(app(x₁, x₂)) = x₁ + x₂
POL(cons) = 0
POL(g) = 0
POL(map_1) = 0
POL(map_2) = 0
POL(map_3) = 1

(46) Obligation:

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

APP(app(app(app(map_3, x0), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, x0), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(app(map_3, map_2), g), x1), app(app(cons, y_1), y_2))
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, y_2), y_3))
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))

R is empty.
The set Q consists of the following terms:

app(app(map_1, x0), app(app(cons, x1), x2))
app(app(app(map_2, x0), x1), app(app(cons, x2), x3))
app(app(app(app(map_3, x0), g), x1), app(app(cons, x2), x3))

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

(47) 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(app(app(map_3, x0), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, x0), g), x1), app(app(cons, y_2), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, map_2), g), x1), app(app(cons, x2), app(app(cons, y_1), y_2))) → APP(app(app(app(map_3, map_2), g), x1), app(app(cons, y_1), y_2))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, x2), app(app(cons, y_2), y_3))) → APP(app(app(app(map_3, app(map_3, y_0)), g), x1), app(app(cons, y_2), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(app(map_3, map_2), g)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, y_0), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(app(map_3, app(map_3, y_0)), g)), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, map_1), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, map_1), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, y_0)), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(map_2, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), y_3)), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_3, y_0), g))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), y_2)), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(map_2, y_1), y_2)), y_3)), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(app(app(map_3, y_1), g), y_2)), y_3)), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, app(map_1, y_1)), y_2)), y_3))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, y_2), y_3))), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(map_2, y_2), y_3)), y_4))), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4))), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))) → APP(app(app(app(map_3, app(map_2, map_1)), g), x1), app(app(cons, app(app(cons, y_1), app(app(cons, app(map_1, y_2)), y_3))), y_4))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, y_0), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(map_2, y_0)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, app(app(map_3, y_0), g)), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(map_2, y_2), y_3)), y_4)), y_5)), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(app(map_3, y_2), g), y_3)), y_4)), y_5)), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(map_1, y_2)), y_3)), y_4)), y_5))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), y_4)), y_5))), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(map_2, y_3), y_4)), y_5)), y_6))), y_7))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(app(map_3, y_3), g), y_4)), y_5)), y_6))), y_7))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(map_2, map_1), y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(map_1, y_3)), y_4)), y_5))), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(app(app(map_3, y_0), g), y_1))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, y_4), y_5)), y_6))), y_7))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, y_0))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6))), y_7))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(map_2, y_0), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6)), y_7))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(app(app(map_3, y_0), g), y_1)))), g), x1), app(app(cons, app(app(cons, y_3), app(app(cons, app(app(cons, app(app(cons, y_4), y_5)), y_6)), y_7))), y_8))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), y_3)), y_4)), y_5)), y_6))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, app(app(cons, app(app(cons, y_2), app(app(cons, y_3), y_4))), y_5)), y_6)), y_7))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), y_4)), y_5)), y_6))), y_7))
The graph contains the following edges 1 >= 1, 2 > 2
APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, x2), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))) → APP(app(app(app(map_3, app(map_2, app(map_1, app(map_1, y_0)))), g), x1), app(app(cons, app(app(cons, y_2), app(app(cons, app(app(cons, app(app(cons, y_3), app(app(cons, y_4), y_5))), y_6)), y_7))), y_8))
The graph contains the following edges 1 >= 1, 2 > 2

(0) Obligation:

(1) Overlay + Local Confluence (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) ForwardInstantiation (EQUIVALENT transformation)

(8) Obligation:

(9) ForwardInstantiation (EQUIVALENT transformation)

(10) Obligation:

(11) ForwardInstantiation (EQUIVALENT transformation)

(12) Obligation:

(13) ForwardInstantiation (EQUIVALENT transformation)

(14) Obligation:

(15) ForwardInstantiation (EQUIVALENT transformation)

(16) Obligation:

(17) ForwardInstantiation (EQUIVALENT transformation)

(18) Obligation:

(19) UsableRulesProof (EQUIVALENT transformation)

(20) Obligation:

(21) ForwardInstantiation (EQUIVALENT transformation)

(22) Obligation:

(23) ForwardInstantiation (EQUIVALENT transformation)

(24) Obligation:

(25) MRRProof (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) Complex Obligation (AND)

(29) Obligation:

(30) ATransformationProof (EQUIVALENT transformation)

(31) Obligation:

(32) QReductionProof (EQUIVALENT transformation)

(33) Obligation:

(34) QDPSizeChangeProof (EQUIVALENT transformation)

(35) TRUE

(36) Obligation:

(37) MRRProof (EQUIVALENT transformation)

(38) Obligation:

(39) DependencyGraphProof (EQUIVALENT transformation)

(40) Complex Obligation (AND)

(41) Obligation:

(42) QDPSizeChangeProof (EQUIVALENT transformation)

(43) TRUE

(44) Obligation:

(45) MRRProof (EQUIVALENT transformation)

(46) Obligation:

(47) QDPSizeChangeProof (EQUIVALENT transformation)

(48) TRUE