Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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.


↳ QTRS
  ↳ DependencyPairsProof

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.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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(f, x)
APP(minsort, app(app(cons, x), y)) → APP(min, x)
APP(app(min, x), app(app(cons, y), z)) → APP(app(min, x), z)
APP(minsort, app(app(cons, x), y)) → APP(del, app(app(min, x), y))
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(eq, app(s, x)), app(s, y)) → APP(eq, x)
APP(app(le, app(s, x)), app(s, y)) → APP(le, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(min, x), app(app(cons, y), z)) → APP(app(min, y), z)
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(minsort, app(app(cons, x), y)) → APP(cons, 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(if, app(app(le, x), y))
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(min, x), app(app(cons, y), z)) → APP(app(if, app(app(le, x), y)), app(app(min, x), z))
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(app(app(filter2, false), f), x), xs) → APP(filter, f)
APP(app(del, x), app(app(cons, y), z)) → APP(app(cons, y), app(app(del, x), z))
APP(minsort, app(app(cons, x), y)) → APP(app(min, x), y)
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)
APP(app(min, x), app(app(cons, y), z)) → APP(app(le, x), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(min, x), app(app(cons, y), z)) → APP(min, y)
APP(app(del, x), app(app(cons, y), z)) → APP(eq, x)
APP(minsort, app(app(cons, x), y)) → APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(app(del, x), app(app(cons, y), z)) → APP(app(if, app(app(eq, x), y)), z)
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
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(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(min, x), app(app(cons, y), z)) → APP(le, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(minsort, app(app(cons, x), y)) → APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(app(del, x), app(app(cons, y), z)) → APP(app(del, x), z)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, 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)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

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(f, x)
APP(minsort, app(app(cons, x), y)) → APP(min, x)
APP(app(min, x), app(app(cons, y), z)) → APP(app(min, x), z)
APP(minsort, app(app(cons, x), y)) → APP(del, app(app(min, x), y))
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(eq, app(s, x)), app(s, y)) → APP(eq, x)
APP(app(le, app(s, x)), app(s, y)) → APP(le, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(min, x), app(app(cons, y), z)) → APP(app(min, y), z)
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(minsort, app(app(cons, x), y)) → APP(cons, 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(if, app(app(le, x), y))
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(min, x), app(app(cons, y), z)) → APP(app(if, app(app(le, x), y)), app(app(min, x), z))
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(app(app(filter2, false), f), x), xs) → APP(filter, f)
APP(app(del, x), app(app(cons, y), z)) → APP(app(cons, y), app(app(del, x), z))
APP(minsort, app(app(cons, x), y)) → APP(app(min, x), y)
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)
APP(app(min, x), app(app(cons, y), z)) → APP(app(le, x), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(min, x), app(app(cons, y), z)) → APP(min, y)
APP(app(del, x), app(app(cons, y), z)) → APP(eq, x)
APP(minsort, app(app(cons, x), y)) → APP(minsort, app(app(del, app(app(min, x), y)), app(app(cons, x), y)))
APP(app(del, x), app(app(cons, y), z)) → APP(app(if, app(app(eq, x), y)), z)
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
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(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(min, x), app(app(cons, y), z)) → APP(le, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(minsort, app(app(cons, x), y)) → APP(app(del, app(app(min, x), y)), app(app(cons, x), y))
APP(app(del, x), app(app(cons, y), z)) → APP(app(del, x), z)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, 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)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 6 SCCs with 29 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


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.
none
Used ordering: Polynomial interpretation [25,35]:

POL(eq) = 0   
POL(APP(x1, x2)) = (3)x_2   
POL(app(x1, x2)) = 2 + (2)x_2   
POL(s) = 0   
The value of delta used in the strict ordering is 6.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


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.
none
Used ordering: Polynomial interpretation [25,35]:

POL(APP(x1, x2)) = (4)x_2   
POL(cons) = 2   
POL(app(x1, x2)) = 1 + (4)x_2   
POL(del) = 0   
The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


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.
none
Used ordering: Polynomial interpretation [25,35]:

POL(APP(x1, x2)) = (4)x_1 + (3)x_2   
POL(app(x1, x2)) = 4 + (2)x_2   
POL(s) = 0   
POL(le) = 0   
The value of delta used in the strict ordering is 44.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

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)

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.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


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)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(APP(x1, x2)) = (4)x_1 + (4)x_2   
POL(cons) = 3   
POL(app(x1, x2)) = (4)x_1 + (4)x_2   
POL(min) = 0   
The value of delta used in the strict ordering is 192.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP

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)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof

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

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

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.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


APP(app(filter, 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(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
The remaining pairs can at least be oriented weakly.

APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
Used ordering: Polynomial interpretation [25,35]:

POL(eq) = 0   
POL(if) = 0   
POL(true) = 0   
POL(s) = 0   
POL(min) = 0   
POL(0) = 0   
POL(filter) = 0   
POL(del) = 0   
POL(APP(x1, x2)) = (1/4)x_2   
POL(minsort) = 0   
POL(cons) = 1/4   
POL(map) = 0   
POL(false) = 0   
POL(app(x1, x2)) = 1/4 + (1/4)x_1 + (4)x_2   
POL(filter2) = 0   
POL(nil) = 0   
POL(le) = 0   
The value of delta used in the strict ordering is 21/256.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof

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)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.