Termination w.r.t. Q proof of /tmp/tmp-E4T01/17.xml

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

    ↳5 QDP

        ↳6 QDPOrderProof (⇔)

        ↳7 QDP

        ↳8 PisEmptyProof (⇔)

        ↳9 TRUE

    ↳10 QDP

        ↳11 QDPOrderProof (⇔)

        ↳12 QDP

        ↳13 PisEmptyProof (⇔)

        ↳14 TRUE

    ↳15 QDP

        ↳16 QDPOrderProof (⇔)

        ↳17 QDP

        ↳18 DependencyGraphProof (⇔)

        ↳19 TRUE

(0) Obligation:

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

app(app(., 1), x) → x
app(app(., x), 1) → x
app(app(., app(i, x)), x) → 1
app(app(., x), app(i, x)) → 1
app(app(., app(i, y)), app(app(., y), z)) → z
app(app(., y), app(app(., app(i, y)), z)) → z
app(app(., app(app(., x), y)), z) → app(app(., x), app(app(., y), z))
app(i, 1) → 1
app(i, app(i, x)) → x
app(i, app(app(., x), y)) → app(app(., app(i, y)), app(i, x))
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(., app(app(., x), y)), z) → APP(app(., x), app(app(., y), z))
APP(app(., app(app(., x), y)), z) → APP(app(., y), z)
APP(app(., app(app(., x), y)), z) → APP(., y)
APP(i, app(app(., x), y)) → APP(app(., app(i, y)), app(i, x))
APP(i, app(app(., x), y)) → APP(., app(i, y))
APP(i, app(app(., x), y)) → APP(i, y)
APP(i, app(app(., x), y)) → APP(i, x)
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(., 1), x) → x
app(app(., x), 1) → x
app(app(., app(i, x)), x) → 1
app(app(., x), app(i, x)) → 1
app(app(., app(i, y)), app(app(., y), z)) → z
app(app(., y), app(app(., app(i, y)), z)) → z
app(app(., app(app(., x), y)), z) → app(app(., x), app(app(., y), z))
app(i, 1) → 1
app(i, app(i, x)) → x
app(i, app(app(., x), y)) → app(app(., app(i, y)), app(i, x))
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 3 SCCs with 12 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

APP(app(., app(app(., x), y)), z) → APP(app(., y), z)
APP(app(., app(app(., x), y)), z) → APP(app(., x), app(app(., y), z))

The TRS R consists of the following rules:

app(app(., 1), x) → x
app(app(., x), 1) → x
app(app(., app(i, x)), x) → 1
app(app(., x), app(i, x)) → 1
app(app(., app(i, y)), app(app(., y), z)) → z
app(app(., y), app(app(., app(i, y)), z)) → z
app(app(., app(app(., x), y)), z) → app(app(., x), app(app(., y), z))
app(i, 1) → 1
app(i, app(i, x)) → x
app(i, app(app(., x), y)) → app(app(., app(i, y)), app(i, x))
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].

The following pairs can be oriented strictly and are deleted.

APP(app(., app(app(., x), y)), z) → APP(app(., y), z)
APP(app(., app(app(., x), y)), z) → APP(app(., x), app(app(., y), z))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x0, x1, x2) = APP(x0, x1)

Tags:
APP has argument tags [1,2,0] and root tag 0

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
APP(x1, x2) = APP(x1)
app(x1, x2) = app(x1, x2)
. = .
1 = 1
i = i

Recursive path order with status [RPO].
Quasi-Precedence:

APP₁ > [app₂, .]

Status:

APP₁: multiset
app₂: multiset
.: multiset
1: multiset
i: multiset

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

app(app(., 1), x) → x
app(app(., x), 1) → x
app(app(., app(i, x)), x) → 1
app(app(., x), app(i, x)) → 1
app(app(., app(i, y)), app(app(., y), z)) → z
app(app(., y), app(app(., app(i, y)), z)) → z
app(app(., app(app(., x), y)), z) → app(app(., x), app(app(., y), z))
app(i, 1) → 1
app(i, app(i, x)) → x
app(i, app(app(., x), y)) → app(app(., app(i, y)), app(i, x))
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) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

APP(i, app(app(., x), y)) → APP(i, x)
APP(i, app(app(., x), y)) → APP(i, y)

The TRS R consists of the following rules:

app(app(., 1), x) → x
app(app(., x), 1) → x
app(app(., app(i, x)), x) → 1
app(app(., x), app(i, x)) → 1
app(app(., app(i, y)), app(app(., y), z)) → z
app(app(., y), app(app(., app(i, y)), z)) → z
app(app(., app(app(., x), y)), z) → app(app(., x), app(app(., y), z))
app(i, 1) → 1
app(i, app(i, x)) → x
app(i, app(app(., x), y)) → app(app(., app(i, y)), app(i, x))
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.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

APP(i, app(app(., x), y)) → APP(i, x)
APP(i, app(app(., x), y)) → APP(i, y)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x0, x1, x2) = APP(x0)

Tags:
APP has argument tags [0,0,3] and root tag 0

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

Recursive path order with status [RPO].
Quasi-Precedence:

[i, .] > APP₁

Status:

APP₁: [1]
i: multiset
app₂: multiset
.: multiset

The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

app(app(., 1), x) → x
app(app(., x), 1) → x
app(app(., app(i, x)), x) → 1
app(app(., x), app(i, x)) → 1
app(app(., app(i, y)), app(app(., y), z)) → z
app(app(., y), app(app(., app(i, y)), z)) → z
app(app(., app(app(., x), y)), z) → app(app(., x), app(app(., y), z))
app(i, 1) → 1
app(i, app(i, x)) → x
app(i, app(app(., x), y)) → app(app(., app(i, y)), app(i, x))
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) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

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

The TRS R consists of the following rules:

app(app(., 1), x) → x
app(app(., x), 1) → x
app(app(., app(i, x)), x) → 1
app(app(., x), app(i, x)) → 1
app(app(., app(i, y)), app(app(., y), z)) → z
app(app(., y), app(app(., app(i, y)), z)) → z
app(app(., app(app(., x), y)), z) → app(app(., x), app(app(., y), z))
app(i, 1) → 1
app(i, app(i, x)) → x
app(i, app(app(., x), y)) → app(app(., app(i, y)), app(i, x))
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.

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
APP(x0, x1, x2) = APP(x0, x2)

Tags:
APP has argument tags [2,1,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
APP(x1, x2) = APP
app(x1, x2) = app(x1, x2)
map = map
cons = cons
filter = filter
filter2 = filter2
true = true
false = false
. = .
1 = 1
i = i
nil = nil

Recursive path order with status [RPO].
Quasi-Precedence:

filter > [APP, cons, filter2] > [app₂, true, .] > [map, i, nil]
false > [app₂, true, .] > [map, i, nil]
1 > [map, i, nil]

Status:

APP: multiset
app₂: multiset
map: multiset
cons: multiset
filter: multiset
filter2: multiset
true: multiset
false: multiset
.: multiset
1: multiset
i: multiset
nil: multiset

The following usable rules [FROCOS05] were oriented: none

(17) 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(., 1), x) → x
app(app(., x), 1) → x
app(app(., app(i, x)), x) → 1
app(app(., x), app(i, x)) → 1
app(app(., app(i, y)), app(app(., y), z)) → z
app(app(., y), app(app(., app(i, y)), z)) → z
app(app(., app(app(., x), y)), z) → app(app(., x), app(app(., y), z))
app(i, 1) → 1
app(i, app(i, x)) → x
app(i, app(app(., x), y)) → app(app(., app(i, y)), app(i, x))
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.

(18) DependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) PisEmptyProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) PisEmptyProof (EQUIVALENT transformation)

(14) TRUE

(15) Obligation:

(16) QDPOrderProof (EQUIVALENT transformation)

(17) Obligation:

(18) DependencyGraphProof (EQUIVALENT transformation)

(19) TRUE