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 QDPOrderProof (⇔)
    ↳29 QDP
    ↳30 DependencyGraphProof (⇔)
    ↳31 TRUE
  ↳32 QDP
  ↳33 QDP
    ↳34 QDPOrderProof (⇔)
    ↳35 QDP
    ↳36 DependencyGraphProof (⇔)
    ↳37 TRUE

(0) Obligation:

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

app'(app'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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'(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'(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'(app, app'(app'(add, n), x)), y) → APP'(app'(add, n), app'(app'(app, x), y))
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app'(app, x), y)
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app, x)
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(if_min, app'(app'(le, n), m))
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(app'(le, n), m)
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(le, n)
APP'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → APP'(min, app'(app'(add, n), x))
APP'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → APP'(app'(add, n), x)
APP'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → APP'(min, app'(app'(add, m), x))
APP'(app'(rm, n), app'(app'(add, m), x)) → APP'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
APP'(app'(rm, n), app'(app'(add, m), x)) → APP'(app'(if_rm, app'(app'(eq, n), m)), n)
APP'(app'(rm, n), app'(app'(add, m), x)) → APP'(if_rm, app'(app'(eq, n), m))
APP'(app'(rm, n), app'(app'(add, m), x)) → APP'(app'(eq, n), m)
APP'(app'(rm, n), app'(app'(add, m), x)) → APP'(eq, n)
APP'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → APP'(app'(rm, n), x)
APP'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → APP'(rm, n)
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → APP'(app'(add, m), app'(app'(rm, n), x))
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → APP'(app'(rm, n), x)
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → APP'(rm, n)
APP'(app'(minsort, app'(app'(add, n), x)), y) → APP'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
APP'(app'(minsort, app'(app'(add, n), x)), y) → APP'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x))
APP'(app'(minsort, app'(app'(add, n), x)), y) → APP'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x))))
APP'(app'(minsort, app'(app'(add, n), x)), y) → APP'(app'(eq, n), app'(min, app'(app'(add, n), x)))
APP'(app'(minsort, app'(app'(add, n), x)), y) → APP'(eq, n)
APP'(app'(minsort, app'(app'(add, n), x)), y) → APP'(min, app'(app'(add, n), x))
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(minsort, app'(app'(app, app'(app'(rm, n), x)), y))
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(app'(app, app'(app'(rm, n), x)), y)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(app, app'(app'(rm, n), x))
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(app'(rm, n), x)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(rm, n)
APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → APP'(app'(minsort, x), app'(app'(add, n), y))
APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → APP'(minsort, x)
APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → APP'(app'(add, n), y)
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(app'(add, app'(f, x)), app'(app'(map, f), xs))
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(add, app'(f, x))
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(f, x)
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(app'(map, f), xs)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(app'(filter2, app'(f, x)), f), x)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(filter2, app'(f, x)), f)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(filter2, app'(f, x))
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(f, x)
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(app'(add, x), app'(app'(filter, f), xs))
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(add, 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'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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 7 SCCs with 37 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app'(app, x), y)

The TRS R consists of the following rules:

app'(app'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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

app1(add(n, x), y) → app1(x, y)

The a-transformed usable rules are
none


The following pairs can be oriented strictly and are deleted.


APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app'(app, x), y)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
[app12, add2]

Status:
add2: multiset
app12: [2,1]


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'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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'(le, app'(s, x)), app'(s, y)) → APP'(app'(le, x), y)

The TRS R consists of the following rules:

app'(app'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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

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: Recursive path order with status [RPO].
Quasi-Precedence:
[le12, s1]

Status:
le12: [1,2]
s1: [1]


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'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
APP'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → APP'(min, app'(app'(add, n), x))
APP'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → APP'(min, app'(app'(add, m), x))

The TRS R consists of the following rules:

app'(app'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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

min1(add(n, add(m, x))) → if_min1(le(n, m), add(n, add(m, x)))
if_min1(true, add(n, add(m, x))) → min1(add(n, x))
if_min1(false, add(n, add(m, x))) → min1(add(m, x))

The a-transformed usable rules are

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)


The following pairs can be oriented strictly and are deleted.


APP'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
APP'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → APP'(min, app'(app'(add, n), x))
APP'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → APP'(min, app'(app'(add, m), x))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
min1(x1)  =  min1(x1)
add(x1, x2)  =  add(x1, x2)
if_min1(x1, x2)  =  if_min1(x1, x2)
le(x1, x2)  =  le(x1)
true  =  true
false  =  false
0  =  0
s(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[add2, true, 0] > [min11, ifmin12, le1, false]

Status:
add2: [2,1]
min11: [1]
ifmin12: [1,2]
true: multiset
le1: multiset
false: multiset
0: multiset


The following usable rules [FROCOS05] were oriented:

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)

(19) Obligation:

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

app'(app'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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'(eq, app'(s, x)), app'(s, y)) → APP'(app'(eq, x), y)

The TRS R consists of the following rules:

app'(app'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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

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: Recursive path order with status [RPO].
Quasi-Precedence:
[eq12, s1]

Status:
s1: [1]
eq12: [1,2]


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'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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'(app'(rm, n), app'(app'(add, m), x)) → APP'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
APP'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → APP'(app'(rm, n), x)
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → APP'(app'(rm, n), x)

The TRS R consists of the following rules:

app'(app'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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) 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

rm1(n, add(m, x)) → if_rm1(eq(n, m), n, add(m, x))
if_rm1(true, n, add(m, x)) → rm1(n, x)
if_rm1(false, n, add(m, x)) → rm1(n, x)

The a-transformed usable rules are

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)


The following pairs can be oriented strictly and are deleted.


APP'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → APP'(app'(rm, n), x)
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → APP'(app'(rm, n), x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
rm1(x1, x2)  =  rm1(x1, x2)
add(x1, x2)  =  add(x2)
if_rm1(x1, x2, x3)  =  if_rm1(x2, x3)
eq(x1, x2)  =  eq(x2)
true  =  true
false  =  false
0  =  0
s(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
eq1 > true > [rm12, add1, ifrm12]
eq1 > false > [rm12, add1, ifrm12]
0 > [rm12, add1, ifrm12]

Status:
add1: [1]
eq1: [1]
ifrm12: multiset
true: multiset
false: multiset
rm12: multiset
0: multiset


The following usable rules [FROCOS05] were oriented:

app'(app'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → false
app'(app'(eq, app'(s, x)), 0) → false
app'(app'(eq, app'(s, x)), app'(s, y)) → app'(app'(eq, x), y)

(29) Obligation:

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

APP'(app'(rm, n), app'(app'(add, m), x)) → APP'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))

The TRS R consists of the following rules:

app'(app'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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.

(30) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(31) TRUE

(32) Obligation:

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

APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil)
APP'(app'(minsort, app'(app'(add, n), x)), y) → APP'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → APP'(app'(minsort, x), app'(app'(add, n), y))

The TRS R consists of the following rules:

app'(app'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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.

(33) Obligation:

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

APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(app'(map, f), xs)
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(f, x)
APP'(app'(filter, f), app'(app'(add, 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'(add, 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'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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.

(34) 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'(add, x), xs)) → APP'(app'(map, f), xs)
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(f, x)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
APP'(app'(filter, f), app'(app'(add, 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
add  =  add
filter  =  filter
filter2  =  filter2
true  =  true
false  =  false
minsort  =  minsort
nil  =  nil
rm  =  rm
if_rm  =  if_rm
eq  =  eq
min  =  min
if_min  =  if_min
le  =  le
if_minsort  =  if_minsort
app  =  app
s  =  s
0  =  0

Recursive path order with status [RPO].
Quasi-Precedence:
filter > APP'1 > filter2 > [add, true, eq, app] > [app'2, minsort, le] > map
filter > APP'1 > filter2 > [add, true, eq, app] > [app'2, minsort, le] > [false, s]
filter > nil
[rm, ifrm] > [add, true, eq, app] > [app'2, minsort, le] > map
[rm, ifrm] > [add, true, eq, app] > [app'2, minsort, le] > [false, s]
[rm, ifrm] > nil
ifmin > min > [add, true, eq, app] > [app'2, minsort, le] > map
ifmin > min > [add, true, eq, app] > [app'2, minsort, le] > [false, s]
ifminsort > [add, true, eq, app] > [app'2, minsort, le] > map
ifminsort > [add, true, eq, app] > [app'2, minsort, le] > [false, s]
ifminsort > nil

Status:
eq: multiset
ifminsort: multiset
rm: multiset
true: multiset
app'2: multiset
s: multiset
min: multiset
0: multiset
filter: multiset
minsort: multiset
add: multiset
app: multiset
map: multiset
APP'1: [1]
false: multiset
filter2: multiset
ifrm: multiset
ifmin: multiset
nil: multiset
le: multiset


The following usable rules [FROCOS05] were oriented: none

(35) 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'(eq, 0), 0) → true
app'(app'(eq, 0), app'(s, x)) → 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'(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'(app, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(min, app'(app'(add, n), nil)) → n
app'(min, app'(app'(add, n), app'(app'(add, m), x))) → app'(app'(if_min, app'(app'(le, n), m)), app'(app'(add, n), app'(app'(add, m), x)))
app'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, n), x))
app'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → app'(min, app'(app'(add, m), x))
app'(app'(rm, n), nil) → nil
app'(app'(rm, n), app'(app'(add, m), x)) → app'(app'(app'(if_rm, app'(app'(eq, n), m)), n), app'(app'(add, m), x))
app'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → app'(app'(rm, n), x)
app'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(rm, n), x))
app'(app'(minsort, nil), nil) → nil
app'(app'(minsort, app'(app'(add, n), x)), y) → app'(app'(app'(if_minsort, app'(app'(eq, n), app'(min, app'(app'(add, n), x)))), app'(app'(add, n), x)), y)
app'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
app'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → app'(app'(minsort, x), app'(app'(add, n), y))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, 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'(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'(le, 0), x0)
app'(app'(le, app'(s, x0)), 0)
app'(app'(le, app'(s, x0)), app'(s, x1))
app'(app'(app, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(min, app'(app'(add, x0), nil))
app'(min, app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, true), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(if_min, false), app'(app'(add, x0), app'(app'(add, x1), x2)))
app'(app'(rm, x0), nil)
app'(app'(rm, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_rm, false), x0), app'(app'(add, x1), x2))
app'(app'(minsort, nil), nil)
app'(app'(minsort, app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, true), app'(app'(add, x0), x1)), x2)
app'(app'(app'(if_minsort, false), app'(app'(add, x0), x1)), x2)
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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.

(36) DependencyGraphProof (EQUIVALENT transformation)

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

(37) TRUE