app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
↳ QTRS
↳ Non-Overlap Check
app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, app2(s, x)), app2(app2(minus, y), x))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(minus, y), x)
APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(minus, x)
APP2(perfectp, app2(s, x)) -> APP2(app2(app2(f, x), app2(s, 0)), app2(s, x))
APP2(perfectp, app2(s, x)) -> APP2(f, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, x), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(le, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(if, app2(app2(le, x), y))
APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(le, x)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(minus, z)
APP2(perfectp, app2(s, x)) -> APP2(app2(f, x), app2(s, 0))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(minus, y)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, x), u), z)
APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(f, x), u)
APP2(perfectp, app2(s, x)) -> APP2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, x), u), z), u)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(f, x)
APP2(perfectp, app2(s, x)) -> APP2(s, 0)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(le, x), y)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x)))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(minus, z), app2(s, x))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(f, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u))
app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, app2(s, x)), app2(app2(minus, y), x))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(minus, y), x)
APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(minus, x)
APP2(perfectp, app2(s, x)) -> APP2(app2(app2(f, x), app2(s, 0)), app2(s, x))
APP2(perfectp, app2(s, x)) -> APP2(f, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(f, x), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(le, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(if, app2(app2(le, x), y))
APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(le, x)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(minus, z)
APP2(perfectp, app2(s, x)) -> APP2(app2(f, x), app2(s, 0))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(minus, y)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, x), u), z)
APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(f, x), u)
APP2(perfectp, app2(s, x)) -> APP2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, x), u), z), u)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(f, x)
APP2(perfectp, app2(s, x)) -> APP2(s, 0)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(le, x), y)
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x)))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(minus, z), app2(s, x))
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(f, x)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u))
app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)
app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)
The following pairs can be strictly oriented and are deleted.
The remaining pairs can at least by weakly be oriented.
APP2(app2(le, app2(s, x)), app2(s, y)) -> APP2(app2(le, x), y)
trivial
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)
app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)
The following pairs can be strictly oriented and are deleted.
The remaining pairs can at least by weakly be oriented.
APP2(app2(minus, app2(s, x)), app2(s, y)) -> APP2(app2(minus, x), y)
trivial
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)
↳ QTRS
↳ Non-Overlap Check
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
APP2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> APP2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)
APP2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> APP2(app2(app2(app2(f, x), u), z), u)
app2(app2(minus, 0), y) -> 0
app2(app2(minus, app2(s, x)), 0) -> app2(s, x)
app2(app2(minus, app2(s, x)), app2(s, y)) -> app2(app2(minus, x), y)
app2(app2(le, 0), y) -> true
app2(app2(le, app2(s, x)), 0) -> false
app2(app2(le, app2(s, x)), app2(s, y)) -> app2(app2(le, x), y)
app2(app2(app2(if, true), x), y) -> x
app2(app2(app2(if, false), x), y) -> y
app2(perfectp, 0) -> false
app2(perfectp, app2(s, x)) -> app2(app2(app2(app2(f, x), app2(s, 0)), app2(s, x)), app2(s, x))
app2(app2(app2(app2(f, 0), y), 0), u) -> true
app2(app2(app2(app2(f, 0), y), app2(s, z)), u) -> false
app2(app2(app2(app2(f, app2(s, x)), 0), z), u) -> app2(app2(app2(app2(f, x), u), app2(app2(minus, z), app2(s, x))), u)
app2(app2(app2(app2(f, app2(s, x)), app2(s, y)), z), u) -> app2(app2(app2(if, app2(app2(le, x), y)), app2(app2(app2(app2(f, app2(s, x)), app2(app2(minus, y), x)), z), u)), app2(app2(app2(app2(f, x), u), z), u))
app2(app2(minus, 0), x0)
app2(app2(minus, app2(s, x0)), 0)
app2(app2(minus, app2(s, x0)), app2(s, x1))
app2(app2(le, 0), x0)
app2(app2(le, app2(s, x0)), 0)
app2(app2(le, app2(s, x0)), app2(s, x1))
app2(app2(app2(if, true), x0), x1)
app2(app2(app2(if, false), x0), x1)
app2(perfectp, 0)
app2(perfectp, app2(s, x0))
app2(app2(app2(app2(f, 0), x0), 0), x1)
app2(app2(app2(app2(f, 0), x0), app2(s, x1)), x2)
app2(app2(app2(app2(f, app2(s, x0)), 0), x1), x2)
app2(app2(app2(app2(f, app2(s, x0)), app2(s, x1)), x2), x3)