Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
  ↳7 QDP
    ↳8 QDPOrderProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 QDPOrderProof (⇔)
    ↳14 QDP
    ↳15 PisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 QDP
    ↳18 QDPOrderProof (⇔)
    ↳19 QDP
    ↳20 PisEmptyProof (⇔)
    ↳21 TRUE
  ↳22 QDP
    ↳23 QDPOrderProof (⇔)
    ↳24 QDP
    ↳25 PisEmptyProof (⇔)
    ↳26 TRUE
  ↳27 QDP
  ↳28 QDP
    ↳29 QDPOrderProof (⇔)
    ↳30 QDP
    ↳31 DependencyGraphProof (⇔)
    ↳32 TRUE

(0) Obligation:

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

app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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) 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(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

(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(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
APP(app(le, app(s, x)), app(s, y)) → APP(le, x)
APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)
APP(app(eq, app(s, x)), app(s, y)) → APP(eq, x)
APP(minsort, app(app(cons, x), y)) → APP(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
APP(minsort, app(app(cons, x), y)) → APP(cons, app(app(min, x), y))
APP(minsort, app(app(cons, x), y)) → APP(app(min, x), y)
APP(minsort, app(app(cons, x), y)) → APP(min, x)
APP(minsort, app(app(cons, x), y)) → APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(minsort, app(app(cons, x), y)) → APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(minsort, app(app(cons, x), y)) → APP(del, app(app(min, x), y))
APP(app(min, x), app(app(cons, y), z)) → APP(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
APP(app(min, x), app(app(cons, y), z)) → APP(app(if, app(app(le, x), y)), app(app(min, x), z))
APP(app(min, x), app(app(cons, y), z)) → APP(if, app(app(le, x), y))
APP(app(min, x), app(app(cons, y), z)) → APP(app(le, x), y)
APP(app(min, x), app(app(cons, y), z)) → APP(le, x)
APP(app(min, x), app(app(cons, y), z)) → APP(app(min, x), z)
APP(app(min, x), app(app(cons, y), z)) → APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) → APP(min, y)
APP(app(del, x), app(app(cons, y), z)) → APP(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
APP(app(del, x), app(app(cons, y), z)) → APP(app(if, app(app(eq, x), y)), z)
APP(app(del, x), app(app(cons, y), z)) → APP(if, app(app(eq, x), y))
APP(app(del, x), app(app(cons, y), z)) → APP(app(eq, x), y)
APP(app(del, x), app(app(cons, y), z)) → APP(eq, x)
APP(app(del, x), app(app(cons, y), z)) → APP(app(cons, y), app(app(del, x), z))
APP(app(del, x), app(app(cons, y), z)) → APP(app(del, x), z)
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(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 29 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)

The TRS R consists of the following rules:

app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

eq1(s(x), s(y)) → eq1(x, y)

The a-transformed usable rules are
none


The following pairs can be oriented strictly and are deleted.


APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
eq1(x1, x2)  =  eq1(x2)
s(x1)  =  s(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[eq11, s1]

Status:
s1: [1]
eq11: multiset


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(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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(del, x), app(app(cons, y), z)) → APP(app(del, x), z)

The TRS R consists of the following rules:

app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

(13) 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

del1(x, cons(y, z)) → del1(x, z)

The a-transformed usable rules are
none


The following pairs can be oriented strictly and are deleted.


APP(app(del, x), app(app(cons, y), z)) → APP(app(del, x), z)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
del1(x1, x2)  =  del1(x2)
cons(x1, x2)  =  cons(x2)

Recursive path order with status [RPO].
Quasi-Precedence:
[del11, cons1]

Status:
cons1: multiset
del11: multiset


The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

(15) PisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE

(17) Obligation:

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

APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)

The TRS R consists of the following rules:

app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

(18) 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

le1(s(x), s(y)) → le1(x, y)

The a-transformed usable rules are
none


The following pairs can be oriented strictly and are deleted.


APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
le1(x1, x2)  =  le1(x2)
s(x1)  =  s(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[le11, s1]

Status:
le11: multiset
s1: [1]


The following usable rules [FROCOS05] were oriented: none

(19) Obligation:

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

app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

(20) PisEmptyProof (EQUIVALENT transformation)

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

(21) TRUE

(22) Obligation:

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

APP(app(min, x), app(app(cons, y), z)) → APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) → APP(app(min, x), z)

The TRS R consists of the following rules:

app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

(23) 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

min1(x, cons(y, z)) → min1(y, z)
min1(x, cons(y, z)) → min1(x, z)

The a-transformed usable rules are
none


The following pairs can be oriented strictly and are deleted.


APP(app(min, x), app(app(cons, y), z)) → APP(app(min, y), z)
APP(app(min, x), app(app(cons, y), z)) → APP(app(min, x), z)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
min1(x1, x2)  =  min1(x2)
cons(x1, x2)  =  cons(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:
[min11, cons2]

Status:
cons2: multiset
min11: [1]


The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

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

app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

(25) PisEmptyProof (EQUIVALENT transformation)

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

(26) TRUE

(27) Obligation:

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

APP(minsort, app(app(cons, x), y)) → APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))

The TRS R consists of the following rules:

app(app(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

(28) 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(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

(29) 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: Combined order from the following AFS and order.
APP(x1, x2)  =  APP(x2)
app(x1, x2)  =  app(x1, x2)
map  =  map
cons  =  cons
filter  =  filter
filter2  =  filter2
true  =  true
false  =  false
le  =  le
0  =  0
eq  =  eq
s  =  s
if  =  if
del  =  del
nil  =  nil
minsort  =  minsort
min  =  min

Recursive path order with status [RPO].
Quasi-Precedence:
map > [APP1, app2, cons, filter, nil] > filter2 > min
map > [APP1, app2, cons, filter, nil] > [false, s] > le > true > min
map > [APP1, app2, cons, filter, nil] > [false, s] > eq > true > min
map > [APP1, app2, cons, filter, nil] > if > min
map > [APP1, app2, cons, filter, nil] > minsort > min
0 > true > min
del > [APP1, app2, cons, filter, nil] > filter2 > min
del > [APP1, app2, cons, filter, nil] > [false, s] > le > true > min
del > [APP1, app2, cons, filter, nil] > [false, s] > eq > true > min
del > [APP1, app2, cons, filter, nil] > if > min
del > [APP1, app2, cons, filter, nil] > minsort > min

Status:
APP1: multiset
eq: multiset
if: multiset
true: multiset
s: multiset
0: multiset
min: multiset
filter: multiset
del: multiset
minsort: multiset
cons: multiset
map: multiset
false: multiset
app2: [1,2]
filter2: multiset
nil: multiset
le: multiset


The following usable rules [FROCOS05] were oriented: none

(30) 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(le, 0), y) → true
app(app(le, app(s, x)), 0) → false
app(app(le, app(s, x)), app(s, y)) → app(app(le, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, y)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(app(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(minsort, nil) → nil
app(minsort, app(app(cons, x), y)) → app(app(cons, app(app(min, x), y)), app(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y))))
app(app(min, x), nil) → x
app(app(min, x), app(app(cons, y), z)) → app(app(app(if, app(app(le, x), y)), app(app(min, x), z)), app(app(min, y), z))
app(app(del, x), nil) → nil
app(app(del, x), app(app(cons, y), z)) → app(app(app(if, app(app(eq, x), y)), z), app(app(cons, y), app(app(del, x), z)))
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)

The set Q consists of the following terms:

app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(app(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(minsort, nil)
app(minsort, app(app(cons, x0), x1))
app(app(min, x0), nil)
app(app(min, x0), app(app(cons, x1), x2))
app(app(del, x0), nil)
app(app(del, x0), app(app(cons, x1), x2))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

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

(31) DependencyGraphProof (EQUIVALENT transformation)

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

(32) TRUE