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)
↳ QTRS
↳ DependencyPairsProof
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)
APP'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → APP'(rm, n)
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(if_min, app'(app'(le, n), m))
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'(map, f), app'(app'(add, x), xs)) → APP'(app'(add, app'(f, x)), app'(app'(map, f), xs))
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'(eq, n), m)
APP'(app'(eq, app'(s, x)), app'(s, y)) → APP'(app'(eq, x), y)
APP'(app'(rm, n), app'(app'(add, m), x)) → APP'(app'(if_rm, app'(app'(eq, n), m)), n)
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'(filter, f), app'(app'(add, x), xs)) → APP'(app'(filter2, app'(f, x)), f)
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, false), app'(app'(add, n), x)), y) → APP'(app'(minsort, x), app'(app'(add, n), y))
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(app'(le, n), m)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(app'(filter2, app'(f, x)), f), x)
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(add, x)
APP'(app'(le, app'(s, x)), app'(s, y)) → APP'(le, x)
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'(app, app'(app'(add, n), x)), y) → APP'(app'(app, x), y)
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_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'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(filter, f)
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'(minsort, app'(app'(add, n), x)), y) → APP'(min, app'(app'(add, n), x))
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(f, 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'(app'(filter, f), app'(app'(add, x), xs)) → APP'(filter2, app'(f, x))
APP'(app'(app'(app'(filter2, false), f), x), xs) → APP'(filter, f)
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app'(add, n), app'(app'(app, x), y))
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(rm, n)
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → APP'(app'(rm, n), x)
APP'(app'(minsort, app'(app'(add, n), x)), y) → APP'(app'(eq, n), app'(min, app'(app'(add, n), x)))
APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → APP'(minsort, x)
APP'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → APP'(app'(add, n), x)
APP'(app'(eq, app'(s, x)), app'(s, y)) → APP'(eq, x)
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(app'(add, x), app'(app'(filter, f), xs))
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app, x)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(app'(rm, n), x)
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → APP'(rm, n)
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(app'(filter, f), xs)
APP'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → APP'(app'(rm, n), x)
APP'(app'(app'(app'(filter2, false), f), x), xs) → APP'(app'(filter, f), xs)
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(app'(map, f), xs)
APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → APP'(app'(add, n), y)
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'(rm, n), app'(app'(add, m), x)) → APP'(eq, n)
APP'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → APP'(min, 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'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(le, n)
APP'(app'(minsort, app'(app'(add, n), x)), y) → APP'(eq, n)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(app, app'(app'(rm, n), x))
APP'(app'(rm, n), app'(app'(add, m), x)) → APP'(if_rm, app'(app'(eq, n), m))
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, 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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
APP'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → APP'(rm, n)
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(if_min, app'(app'(le, n), m))
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'(map, f), app'(app'(add, x), xs)) → APP'(app'(add, app'(f, x)), app'(app'(map, f), xs))
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'(eq, n), m)
APP'(app'(eq, app'(s, x)), app'(s, y)) → APP'(app'(eq, x), y)
APP'(app'(rm, n), app'(app'(add, m), x)) → APP'(app'(if_rm, app'(app'(eq, n), m)), n)
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'(filter, f), app'(app'(add, x), xs)) → APP'(app'(filter2, app'(f, x)), f)
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, false), app'(app'(add, n), x)), y) → APP'(app'(minsort, x), app'(app'(add, n), y))
APP'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(app'(le, n), m)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(app'(filter2, app'(f, x)), f), x)
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(add, x)
APP'(app'(le, app'(s, x)), app'(s, y)) → APP'(le, x)
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'(app, app'(app'(add, n), x)), y) → APP'(app'(app, x), y)
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_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'(add, n), app'(app'(minsort, app'(app'(app, app'(app'(rm, n), x)), y)), nil))
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(filter, f)
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'(minsort, app'(app'(add, n), x)), y) → APP'(min, app'(app'(add, n), x))
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(f, 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'(app'(filter, f), app'(app'(add, x), xs)) → APP'(filter2, app'(f, x))
APP'(app'(app'(app'(filter2, false), f), x), xs) → APP'(filter, f)
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app'(add, n), app'(app'(app, x), y))
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(rm, n)
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → APP'(app'(rm, n), x)
APP'(app'(minsort, app'(app'(add, n), x)), y) → APP'(app'(eq, n), app'(min, app'(app'(add, n), x)))
APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → APP'(minsort, x)
APP'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → APP'(app'(add, n), x)
APP'(app'(eq, app'(s, x)), app'(s, y)) → APP'(eq, x)
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(app'(add, x), app'(app'(filter, f), xs))
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app, x)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(app'(rm, n), x)
APP'(app'(app'(if_rm, false), n), app'(app'(add, m), x)) → APP'(rm, n)
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(app'(filter, f), xs)
APP'(app'(app'(if_rm, true), n), app'(app'(add, m), x)) → APP'(app'(rm, n), x)
APP'(app'(app'(app'(filter2, false), f), x), xs) → APP'(app'(filter, f), xs)
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(app'(map, f), xs)
APP'(app'(app'(if_minsort, false), app'(app'(add, n), x)), y) → APP'(app'(add, n), y)
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'(rm, n), app'(app'(add, m), x)) → APP'(eq, n)
APP'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → APP'(min, 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'(min, app'(app'(add, n), app'(app'(add, m), x))) → APP'(le, n)
APP'(app'(minsort, app'(app'(add, n), x)), y) → APP'(eq, n)
APP'(app'(app'(if_minsort, true), app'(app'(add, n), x)), y) → APP'(app, app'(app'(rm, n), x))
APP'(app'(rm, n), app'(app'(add, m), x)) → APP'(if_rm, app'(app'(eq, n), m))
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, 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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app'(app, x), y)
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 following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app'(app, x), y)
The value of delta used in the strict ordering is 12.
POL(add) = 1
POL(app) = 0
POL(app'(x1, x2)) = x_1 + (3)x_2
POL(APP'(x1, x2)) = (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
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, 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 following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP'(app'(le, app'(s, x)), app'(s, y)) → APP'(app'(le, x), y)
The value of delta used in the strict ordering is 44.
POL(app'(x1, x2)) = 4 + (2)x_2
POL(s) = 0
POL(APP'(x1, x2)) = (4)x_1 + (3)x_2
POL(le) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
APP'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → APP'(min, app'(app'(add, m), 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'(app'(if_min, true), app'(app'(add, n), app'(app'(add, m), x))) → APP'(min, app'(app'(add, n), x))
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 following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP'(app'(if_min, false), app'(app'(add, n), app'(app'(add, m), x))) → APP'(min, 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))
Used ordering: Polynomial interpretation [25,35]:
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)))
The value of delta used in the strict ordering is 4.
POL(add) = 3
POL(true) = 1
POL(app'(x1, x2)) = 2 + x_2
POL(false) = 4
POL(s) = 13/4
POL(if_min) = 7/2
POL(min) = 4
POL(0) = 9/4
POL(APP'(x1, x2)) = (2)x_2
POL(le) = 3/2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
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'(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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
APP'(app'(eq, app'(s, x)), app'(s, y)) → APP'(app'(eq, x), y)
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 following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP'(app'(eq, app'(s, x)), app'(s, y)) → APP'(app'(eq, x), y)
The value of delta used in the strict ordering is 6.
POL(eq) = 0
POL(app'(x1, x2)) = 2 + (2)x_2
POL(s) = 0
POL(APP'(x1, x2)) = (3)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
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)
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 following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
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)
Used ordering: Polynomial interpretation [25,35]:
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 value of delta used in the strict ordering is 10.
POL(eq) = 0
POL(add) = 0
POL(rm) = 0
POL(true) = 7/4
POL(app'(x1, x2)) = 1/2 + (4)x_1 + x_2
POL(false) = 4
POL(s) = 0
POL(if_rm) = 4
POL(0) = 7/2
POL(APP'(x1, x2)) = (4)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
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'(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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
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))
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'(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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
APP'(app'(filter, 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'(app'(app'(filter2, false), f), x), xs) → APP'(app'(filter, f), xs)
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'(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 following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP'(app'(filter, 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'(map, f), app'(app'(add, x), xs)) → APP'(f, x)
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(app'(map, f), xs)
Used ordering: Polynomial interpretation [25,35]:
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 value of delta used in the strict ordering is 12.
POL(eq) = 0
POL(if_minsort) = 0
POL(rm) = 0
POL(true) = 0
POL(app'(x1, x2)) = 2 + (1/2)x_1 + (4)x_2
POL(s) = 0
POL(0) = 0
POL(min) = 0
POL(APP'(x1, x2)) = (4)x_2
POL(filter) = 0
POL(minsort) = 0
POL(add) = 0
POL(app) = 0
POL(map) = 0
POL(false) = 0
POL(if_rm) = 0
POL(filter2) = 0
POL(if_min) = 0
POL(nil) = 0
POL(le) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
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)
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)