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

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

(0) Obligation:

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

app(app(ack, 0), y) → app(succ, y)
app(app(ack, app(succ, x)), y) → app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) → app(app(ack, x), app(app(ack, app(succ, x)), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

APP(app(ack, 0), y) → APP(succ, y)
APP(app(ack, app(succ, x)), y) → APP(app(ack, x), app(succ, 0))
APP(app(ack, app(succ, x)), y) → APP(ack, x)
APP(app(ack, app(succ, x)), y) → APP(succ, 0)
APP(app(ack, app(succ, x)), app(succ, y)) → APP(app(ack, x), app(app(ack, app(succ, x)), y))
APP(app(ack, app(succ, x)), app(succ, y)) → APP(ack, x)
APP(app(ack, app(succ, x)), app(succ, y)) → APP(app(ack, app(succ, x)), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)

The TRS R consists of the following rules:

app(app(ack, 0), y) → app(succ, y)
app(app(ack, app(succ, x)), y) → app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) → app(app(ack, x), app(app(ack, app(succ, x)), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 14 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

APP(app(ack, app(succ, x)), app(succ, y)) → APP(app(ack, x), app(app(ack, app(succ, x)), y))
APP(app(ack, app(succ, x)), y) → APP(app(ack, x), app(succ, 0))
APP(app(ack, app(succ, x)), app(succ, y)) → APP(app(ack, app(succ, x)), y)

The TRS R consists of the following rules:

app(app(ack, 0), y) → app(succ, y)
app(app(ack, app(succ, x)), y) → app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) → app(app(ack, x), app(app(ack, app(succ, x)), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04]. Here, we combined the reduction pair processor with the A-transformation [FROCOS05] which results in the following intermediate Q-DP Problem.
The a-transformed P is

ack1(succ(x), succ(y)) → ack1(x, ack(succ(x), y))
ack1(succ(x), y) → ack1(x, succ(0))
ack1(succ(x), succ(y)) → ack1(succ(x), y)

The a-transformed usable rules are

ack(succ(x), succ(y)) → ack(x, ack(succ(x), y))
ack(succ(x), y) → ack(x, succ(0))
ack(0, y) → succ(y)


The following pairs can be oriented strictly and are deleted.


APP(app(ack, app(succ, x)), app(succ, y)) → APP(app(ack, x), app(app(ack, app(succ, x)), y))
APP(app(ack, app(succ, x)), y) → APP(app(ack, x), app(succ, 0))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ack1(x1, x2)  =  x1
succ(x1)  =  succ(x1)
ack(x1, x2)  =  ack(x1, x2)
0  =  0

Lexicographic Path Order [LPO].
Precedence:
ack2 > succ1
ack2 > 0

The following usable rules [FROCOS05] were oriented:

app(app(ack, app(succ, x)), app(succ, y)) → app(app(ack, x), app(app(ack, app(succ, x)), y))
app(app(ack, app(succ, x)), y) → app(app(ack, x), app(succ, 0))
app(app(ack, 0), y) → app(succ, y)

(7) Obligation:

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

APP(app(ack, app(succ, x)), app(succ, y)) → APP(app(ack, app(succ, x)), y)

The TRS R consists of the following rules:

app(app(ack, 0), y) → app(succ, y)
app(app(ack, app(succ, x)), y) → app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) → app(app(ack, x), app(app(ack, app(succ, x)), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04]. Here, we combined the reduction pair processor with the A-transformation [FROCOS05] which results in the following intermediate Q-DP Problem.
The a-transformed P is

ack1(succ(x), succ(y)) → ack1(succ(x), y)

The a-transformed usable rules are
none


The following pairs can be oriented strictly and are deleted.


APP(app(ack, app(succ, x)), app(succ, y)) → APP(app(ack, app(succ, x)), y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ack1(x1, x2)  =  x2
succ(x1)  =  succ(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

app(app(ack, 0), y) → app(succ, y)
app(app(ack, app(succ, x)), y) → app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) → app(app(ack, x), app(app(ack, app(succ, x)), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

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

APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)

The TRS R consists of the following rules:

app(app(ack, 0), y) → app(succ, y)
app(app(ack, app(succ, x)), y) → app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) → app(app(ack, x), app(app(ack, app(succ, x)), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
APP(x1, x2)  =  x2
app(x1, x2)  =  app(x1, x2)
cons  =  cons

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)

The TRS R consists of the following rules:

app(app(ack, 0), y) → app(succ, y)
app(app(ack, app(succ, x)), y) → app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) → app(app(ack, x), app(app(ack, app(succ, x)), y))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

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

(15) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(16) TRUE