app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
↳ QTRS
↳ DependencyPairsProof
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
APP(app(gtr, app(s, x)), app(s, y)) → APP(gtr, x)
APP(app(d, app(s, x)), app(s, y)) → APP(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
APP(app(filter, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(sub, app(s, x)), app(s, y)) → APP(app(sub, x), y)
APP(app(filter, p), app(app(cons, x), xs)) → APP(app(filter, p), xs)
APP(app(d, app(s, x)), app(s, y)) → APP(app(sub, y), x)
APP(app(filter, p), app(app(cons, x), xs)) → APP(app(cons, x), app(app(filter, p), xs))
APP(app(d, app(s, x)), app(s, y)) → APP(app(d, app(s, x)), app(app(sub, y), x))
APP(len, app(app(cons, x), xs)) → APP(s, app(len, xs))
APP(app(d, app(s, x)), app(s, y)) → APP(app(if, app(app(gtr, x), y)), false)
APP(app(filter, p), app(app(cons, x), xs)) → APP(if, app(p, x))
APP(app(d, app(s, x)), app(s, y)) → APP(if, app(app(gtr, x), y))
APP(app(d, app(s, x)), app(s, y)) → APP(app(gtr, x), y)
APP(app(filter, p), app(app(cons, x), xs)) → APP(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs)))
APP(app(d, app(s, x)), app(s, y)) → APP(sub, y)
APP(len, app(app(cons, x), xs)) → APP(len, xs)
APP(app(sub, app(s, x)), app(s, y)) → APP(sub, x)
APP(app(d, app(s, x)), app(s, y)) → APP(gtr, x)
APP(app(filter, p), app(app(cons, x), xs)) → APP(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
APP(app(gtr, app(s, x)), app(s, y)) → APP(app(gtr, x), y)
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
APP(app(gtr, app(s, x)), app(s, y)) → APP(gtr, x)
APP(app(d, app(s, x)), app(s, y)) → APP(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
APP(app(filter, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(sub, app(s, x)), app(s, y)) → APP(app(sub, x), y)
APP(app(filter, p), app(app(cons, x), xs)) → APP(app(filter, p), xs)
APP(app(d, app(s, x)), app(s, y)) → APP(app(sub, y), x)
APP(app(filter, p), app(app(cons, x), xs)) → APP(app(cons, x), app(app(filter, p), xs))
APP(app(d, app(s, x)), app(s, y)) → APP(app(d, app(s, x)), app(app(sub, y), x))
APP(len, app(app(cons, x), xs)) → APP(s, app(len, xs))
APP(app(d, app(s, x)), app(s, y)) → APP(app(if, app(app(gtr, x), y)), false)
APP(app(filter, p), app(app(cons, x), xs)) → APP(if, app(p, x))
APP(app(d, app(s, x)), app(s, y)) → APP(if, app(app(gtr, x), y))
APP(app(d, app(s, x)), app(s, y)) → APP(app(gtr, x), y)
APP(app(filter, p), app(app(cons, x), xs)) → APP(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs)))
APP(app(d, app(s, x)), app(s, y)) → APP(sub, y)
APP(len, app(app(cons, x), xs)) → APP(len, xs)
APP(app(sub, app(s, x)), app(s, y)) → APP(sub, x)
APP(app(d, app(s, x)), app(s, y)) → APP(gtr, x)
APP(app(filter, p), app(app(cons, x), xs)) → APP(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
APP(app(gtr, app(s, x)), app(s, y)) → APP(app(gtr, x), y)
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
APP(len, app(app(cons, x), xs)) → APP(len, xs)
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(len, app(app(cons, x), xs)) → APP(len, xs)
The value of delta used in the strict ordering is 7/128.
POL(APP(x1, x2)) = (7/2)x_2
POL(cons) = 1/4
POL(len) = 0
POL(app(x1, x2)) = (1/4)x_1 + (7/2)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
APP(app(gtr, app(s, x)), app(s, y)) → APP(app(gtr, x), y)
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(gtr, app(s, x)), app(s, y)) → APP(app(gtr, x), y)
The value of delta used in the strict ordering is 15/64.
POL(APP(x1, x2)) = (15/4)x_2
POL(app(x1, x2)) = (1/4)x_1 + (13/4)x_2
POL(s) = 1/4
POL(gtr) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
APP(app(sub, app(s, x)), app(s, y)) → APP(app(sub, x), y)
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(sub, app(s, x)), app(s, y)) → APP(app(sub, x), y)
The value of delta used in the strict ordering is 64.
POL(APP(x1, x2)) = (4)x_2
POL(sub) = 0
POL(app(x1, x2)) = (4)x_1 + (2)x_2
POL(s) = 4
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
APP(app(d, app(s, x)), app(s, y)) → APP(app(d, app(s, x)), app(app(sub, y), x))
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
APP(app(filter, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(filter, p), app(app(cons, x), xs)) → APP(app(filter, p), xs)
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(filter, p), app(app(cons, x), xs)) → APP(p, x)
APP(app(filter, p), app(app(cons, x), xs)) → APP(app(filter, p), xs)
The value of delta used in the strict ordering is 21/256.
POL(APP(x1, x2)) = (1/2)x_1 + (1/4)x_2
POL(cons) = 1/4
POL(app(x1, x2)) = 1/4 + (1/4)x_1 + (4)x_2
POL(filter) = 5/4
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
app(app(app(if, true), xs), ys) → xs
app(app(app(if, false), xs), ys) → ys
app(app(sub, x), 0) → x
app(app(sub, app(s, x)), app(s, y)) → app(app(sub, x), y)
app(app(gtr, 0), y) → false
app(app(gtr, app(s, x)), 0) → true
app(app(gtr, app(s, x)), app(s, y)) → app(app(gtr, x), y)
app(app(d, x), 0) → true
app(app(d, app(s, x)), app(s, y)) → app(app(app(if, app(app(gtr, x), y)), false), app(app(d, app(s, x)), app(app(sub, y), x)))
app(len, nil) → 0
app(len, app(app(cons, x), xs)) → app(s, app(len, xs))
app(app(filter, p), nil) → nil
app(app(filter, p), app(app(cons, x), xs)) → app(app(app(if, app(p, x)), app(app(cons, x), app(app(filter, p), xs))), app(app(filter, p), xs))