app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
↳ QTRS
↳ Overlay + Local Confluence
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(f, app(s, x)), app(app(minus, y), x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(minus, y), x)
APP(app(minus, app(s, x)), app(s, y)) → APP(minus, x)
APP(perfectp, app(s, x)) → APP(app(app(f, x), app(s, 0)), app(s, x))
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(cons, app(fun, x)), app(app(map, fun), xs))
APP(perfectp, app(s, x)) → APP(f, x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(f, x), u)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(filter2, app(fun, x)), fun), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(le, x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(fun, x)), fun), x), xs)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(if, app(app(le, x), y))
APP(app(le, app(s, x)), app(s, y)) → APP(le, x)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(minus, z)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(cons, x), app(app(filter, fun), xs))
APP(perfectp, app(s, x)) → APP(app(f, x), app(s, 0))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(minus, y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(f, x), u), z)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(filter2, app(fun, x))
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(minus, app(s, x)), app(s, y)) → APP(app(minus, x), y)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(f, x), u)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(filter2, app(fun, x)), fun)
APP(app(app(app(filter2, false), fun), x), xs) → APP(filter, fun)
APP(perfectp, app(s, x)) → APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(f, x)
APP(perfectp, app(s, x)) → APP(s, 0)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(app(f, x), u), app(app(minus, z), app(s, x)))
APP(app(app(app(filter2, true), fun), x), xs) → APP(filter, fun)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(minus, z), app(s, x))
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(f, x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))
APP(app(app(app(filter2, true), fun), x), xs) → APP(cons, x)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(map, fun), app(app(cons, x), xs)) → APP(cons, app(fun, x))
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(f, app(s, x)), app(app(minus, y), x))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(minus, y), x)
APP(app(minus, app(s, x)), app(s, y)) → APP(minus, x)
APP(perfectp, app(s, x)) → APP(app(app(f, x), app(s, 0)), app(s, x))
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(cons, app(fun, x)), app(app(map, fun), xs))
APP(perfectp, app(s, x)) → APP(f, x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(f, x), u)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(filter2, app(fun, x)), fun), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(le, x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(fun, x)), fun), x), xs)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(if, app(app(le, x), y))
APP(app(le, app(s, x)), app(s, y)) → APP(le, x)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(minus, z)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(cons, x), app(app(filter, fun), xs))
APP(perfectp, app(s, x)) → APP(app(f, x), app(s, 0))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(minus, y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(f, x), u), z)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(filter2, app(fun, x))
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(minus, app(s, x)), app(s, y)) → APP(app(minus, x), y)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(f, x), u)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(filter2, app(fun, x)), fun)
APP(app(app(app(filter2, false), fun), x), xs) → APP(filter, fun)
APP(perfectp, app(s, x)) → APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(f, x)
APP(perfectp, app(s, x)) → APP(s, 0)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(app(f, x), u), app(app(minus, z), app(s, x)))
APP(app(app(app(filter2, true), fun), x), xs) → APP(filter, fun)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(minus, z), app(s, x))
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(f, x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))
APP(app(app(app(filter2, true), fun), x), xs) → APP(cons, x)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(map, fun), app(app(cons, x), xs)) → APP(cons, app(fun, x))
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(minus, y), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(f, app(s, x)), app(app(minus, y), x))
APP(app(minus, app(s, x)), app(s, y)) → APP(minus, x)
APP(perfectp, app(s, x)) → APP(app(app(f, x), app(s, 0)), app(s, x))
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(cons, app(fun, x)), app(app(map, fun), xs))
APP(perfectp, app(s, x)) → APP(f, x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(f, x), u)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(filter2, app(fun, x)), fun), x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(le, x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(if, app(app(le, x), y))
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(fun, x)), fun), x), xs)
APP(app(le, app(s, x)), app(s, y)) → APP(le, x)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(minus, z)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(cons, x), app(app(filter, fun), xs))
APP(perfectp, app(s, x)) → APP(app(f, x), app(s, 0))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(minus, y)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(f, x), u), z)
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(filter2, app(fun, x))
APP(app(minus, app(s, x)), app(s, y)) → APP(app(minus, x), y)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(f, x), u)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(filter2, app(fun, x)), fun)
APP(perfectp, app(s, x)) → APP(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
APP(app(app(app(filter2, false), fun), x), xs) → APP(filter, fun)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(f, app(s, x)), app(app(minus, y), x)), z)
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(perfectp, app(s, x)) → APP(s, 0)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(f, x)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, x), u), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(le, x), y)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(app(f, x), u), app(app(minus, z), app(s, x)))
APP(app(app(app(filter2, true), fun), x), xs) → APP(filter, fun)
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(minus, z), app(s, x))
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(f, x)
APP(app(app(app(filter2, true), fun), x), xs) → APP(cons, x)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(map, fun), app(app(cons, x), xs)) → APP(cons, app(fun, x))
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
LE(s(x), s(y)) → LE(x, y)
minus(0, x0)
minus(s(x0), 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
perfectp(0)
perfectp(s(x0))
f(0, x0, 0, x1)
f(0, x0, s(x1), x2)
f(s(x0), 0, x1, x2)
f(s(x0), s(x1), x2, x3)
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
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)
[LE1, s1]
LE1: multiset
s1: multiset
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
APP(app(minus, app(s, x)), app(s, y)) → APP(app(minus, x), y)
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
MINUS(s(x), s(y)) → MINUS(x, y)
minus(0, x0)
minus(s(x0), 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
perfectp(0)
perfectp(s(x0))
f(0, x0, 0, x1)
f(0, x0, s(x1), x2)
f(s(x0), 0, x1, x2)
f(s(x0), s(x1), x2, x3)
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(minus, app(s, x)), app(s, y)) → APP(app(minus, x), y)
[MINUS1, s1]
MINUS1: multiset
s1: multiset
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, x), u), z), u)
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
F(s(x), s(y), z, u) → F(s(x), minus(y, x), z, u)
F(s(x), s(y), z, u) → F(x, u, z, u)
F(s(x), 0, z, u) → F(x, u, minus(z, s(x)), u)
minus(0, y) → 0
minus(s(x), s(y)) → minus(x, y)
minus(s(x), 0) → s(x)
minus(0, x0)
minus(s(x0), 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
perfectp(0)
perfectp(s(x0))
f(0, x0, 0, x1)
f(0, x0, s(x1), x2)
f(s(x0), 0, x1, x2)
f(s(x0), s(x1), x2, x3)
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(app(app(f, app(s, x)), 0), z), u) → APP(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, x), u), z), u)
Used ordering: Combined order from the following AFS and order.
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
s1 > minus1 > 0
minus1: multiset
s1: multiset
0: multiset
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
F(s(x), s(y), z, u) → F(s(x), minus(y, x), z, u)
minus(0, y) → 0
minus(s(x), 0) → s(x)
minus(s(x), s(y)) → minus(x, y)
minus(0, x0)
minus(s(x0), 0)
minus(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
if(true, x0, x1)
if(false, x0, x1)
perfectp(0)
perfectp(s(x0))
f(0, x0, 0, x1)
f(0, x0, s(x1), x2)
f(s(x0), 0, x1, x2)
f(s(x0), s(x1), x2, x3)
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(app(app(f, app(s, x)), app(s, y)), z), u) → APP(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
s1 > [minus1, 0]
minus1: multiset
s1: [1]
0: multiset
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, x), y)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(fun, x)), fun), x), xs)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(fun, x)), fun), x), xs)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
Used ordering: Combined order from the following AFS and order.
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
map > cons > [app2, filter2, filter, false, minus]
s > [0, f] > true > [app2, filter2, filter, false, minus]
s > le > true > [app2, filter2, filter, false, minus]
nil > [app2, filter2, filter, false, minus]
perfectp > [0, f] > true > [app2, filter2, filter, false, minus]
if > [app2, filter2, filter, false, minus]
map: multiset
0: multiset
perfectp: multiset
s: multiset
minus: multiset
nil: multiset
cons: multiset
filter: multiset
true: multiset
false: multiset
if: multiset
app2: [2,1]
le: multiset
f: multiset
filter2: multiset
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
app(app(minus, 0), y) → 0
app(app(minus, app(s, x)), 0) → app(s, x)
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, 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(if, true), x), y) → x
app(app(app(if, false), x), y) → y
app(perfectp, 0) → false
app(perfectp, app(s, x)) → app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u) → true
app(app(app(app(f, 0), y), app(s, z)), u) → false
app(app(app(app(f, app(s, x)), 0), z), u) → app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)
app(app(app(app(f, app(s, x)), app(s, y)), z), u) → app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil) → nil
app(app(map, fun), app(app(cons, x), xs)) → app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil) → nil
app(app(filter, fun), app(app(cons, x), xs)) → app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs) → app(app(cons, x), app(app(filter, fun), xs))
app(app(app(app(filter2, false), fun), x), xs) → app(app(filter, fun), xs)
app(app(minus, 0), x0)
app(app(minus, app(s, x0)), 0)
app(app(minus, 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(if, true), x0), x1)
app(app(app(if, false), x0), x1)
app(perfectp, 0)
app(perfectp, app(s, x0))
app(app(app(app(f, 0), x0), 0), x1)
app(app(app(app(f, 0), x0), app(s, x1)), x2)
app(app(app(app(f, app(s, x0)), 0), x1), x2)
app(app(app(app(f, app(s, x0)), app(s, x1)), x2), x3)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)