0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
↳7 QDP
↳8 UsableRulesProof (⇔)
↳9 QDP
↳10 ATransformationProof (⇔)
↳11 QDP
↳12 QReductionProof (⇔)
↳13 QDP
↳14 QDPSizeChangeProof (⇔)
↳15 TRUE
↳16 QDP
↳17 UsableRulesProof (⇔)
↳18 QDP
↳19 ATransformationProof (⇔)
↳20 QDP
↳21 QReductionProof (⇔)
↳22 QDP
↳23 QDPOrderProof (⇔)
↳24 QDP
↳25 DependencyGraphProof (⇔)
↳26 TRUE
↳27 QDP
↳28 UsableRulesProof (⇔)
↳29 QDP
↳30 QDPSizeChangeProof (⇔)
↳31 TRUE
app(rev, nil) → nil
app(rev, app(app(cons, x), l)) → app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) → 0
app(app(rev1, app(s, x)), nil) → app(s, x)
app(app(rev1, x), app(app(cons, y), l)) → app(app(rev1, y), l)
app(app(rev2, x), nil) → nil
app(app(rev2, x), app(app(cons, y), l)) → app(rev, app(app(cons, x), app(app(rev2, y), l)))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
app(rev, nil) → nil
app(rev, app(app(cons, x), l)) → app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) → 0
app(app(rev1, app(s, x)), nil) → app(s, x)
app(app(rev1, x), app(app(cons, y), l)) → app(app(rev1, y), l)
app(app(rev2, x), nil) → nil
app(app(rev2, x), app(app(cons, y), l)) → app(rev, app(app(cons, x), app(app(rev2, y), l)))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
app(rev, nil)
app(rev, app(app(cons, x0), x1))
app(app(rev1, 0), nil)
app(app(rev1, app(s, x0)), nil)
app(app(rev1, x0), app(app(cons, x1), x2))
app(app(rev2, x0), nil)
app(app(rev2, x0), app(app(cons, x1), x2))
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(rev, app(app(cons, x), l)) → APP(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
APP(rev, app(app(cons, x), l)) → APP(cons, app(app(rev1, x), l))
APP(rev, app(app(cons, x), l)) → APP(app(rev1, x), l)
APP(rev, app(app(cons, x), l)) → APP(rev1, x)
APP(rev, app(app(cons, x), l)) → APP(app(rev2, x), l)
APP(rev, app(app(cons, x), l)) → APP(rev2, x)
APP(app(rev1, x), app(app(cons, y), l)) → APP(app(rev1, y), l)
APP(app(rev1, x), app(app(cons, y), l)) → APP(rev1, y)
APP(app(rev2, x), app(app(cons, y), l)) → APP(rev, app(app(cons, x), app(app(rev2, y), l)))
APP(app(rev2, x), app(app(cons, y), l)) → APP(app(cons, x), app(app(rev2, y), l))
APP(app(rev2, x), app(app(cons, y), l)) → APP(cons, x)
APP(app(rev2, x), app(app(cons, y), l)) → APP(app(rev2, y), l)
APP(app(rev2, x), app(app(cons, y), l)) → APP(rev2, y)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)
app(rev, nil) → nil
app(rev, app(app(cons, x), l)) → app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) → 0
app(app(rev1, app(s, x)), nil) → app(s, x)
app(app(rev1, x), app(app(cons, y), l)) → app(app(rev1, y), l)
app(app(rev2, x), nil) → nil
app(app(rev2, x), app(app(cons, y), l)) → app(rev, app(app(cons, x), app(app(rev2, y), l)))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
app(rev, nil)
app(rev, app(app(cons, x0), x1))
app(app(rev1, 0), nil)
app(app(rev1, app(s, x0)), nil)
app(app(rev1, x0), app(app(cons, x1), x2))
app(app(rev2, x0), nil)
app(app(rev2, x0), app(app(cons, x1), x2))
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(rev1, x), app(app(cons, y), l)) → APP(app(rev1, y), l)
app(rev, nil) → nil
app(rev, app(app(cons, x), l)) → app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) → 0
app(app(rev1, app(s, x)), nil) → app(s, x)
app(app(rev1, x), app(app(cons, y), l)) → app(app(rev1, y), l)
app(app(rev2, x), nil) → nil
app(app(rev2, x), app(app(cons, y), l)) → app(rev, app(app(cons, x), app(app(rev2, y), l)))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
app(rev, nil)
app(rev, app(app(cons, x0), x1))
app(app(rev1, 0), nil)
app(app(rev1, app(s, x0)), nil)
app(app(rev1, x0), app(app(cons, x1), x2))
app(app(rev2, x0), nil)
app(app(rev2, x0), app(app(cons, x1), x2))
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(rev1, x), app(app(cons, y), l)) → APP(app(rev1, y), l)
app(rev, nil)
app(rev, app(app(cons, x0), x1))
app(app(rev1, 0), nil)
app(app(rev1, app(s, x0)), nil)
app(app(rev1, x0), app(app(cons, x1), x2))
app(app(rev2, x0), nil)
app(app(rev2, x0), app(app(cons, x1), x2))
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)
rev11(x, cons(y, l)) → rev11(y, l)
rev(nil)
rev(cons(x0, x1))
rev1(0, nil)
rev1(s(x0), nil)
rev1(x0, cons(x1, x2))
rev2(x0, nil)
rev2(x0, cons(x1, x2))
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)
rev(nil)
rev(cons(x0, x1))
rev1(0, nil)
rev1(s(x0), nil)
rev1(x0, cons(x1, x2))
rev2(x0, nil)
rev2(x0, cons(x1, x2))
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)
rev11(x, cons(y, l)) → rev11(y, l)
From the DPs we obtained the following set of size-change graphs:
APP(rev, app(app(cons, x), l)) → APP(app(rev2, x), l)
APP(app(rev2, x), app(app(cons, y), l)) → APP(rev, app(app(cons, x), app(app(rev2, y), l)))
APP(app(rev2, x), app(app(cons, y), l)) → APP(app(rev2, y), l)
app(rev, nil) → nil
app(rev, app(app(cons, x), l)) → app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) → 0
app(app(rev1, app(s, x)), nil) → app(s, x)
app(app(rev1, x), app(app(cons, y), l)) → app(app(rev1, y), l)
app(app(rev2, x), nil) → nil
app(app(rev2, x), app(app(cons, y), l)) → app(rev, app(app(cons, x), app(app(rev2, y), l)))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
app(rev, nil)
app(rev, app(app(cons, x0), x1))
app(app(rev1, 0), nil)
app(app(rev1, app(s, x0)), nil)
app(app(rev1, x0), app(app(cons, x1), x2))
app(app(rev2, x0), nil)
app(app(rev2, x0), app(app(cons, x1), x2))
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(rev, app(app(cons, x), l)) → APP(app(rev2, x), l)
APP(app(rev2, x), app(app(cons, y), l)) → APP(rev, app(app(cons, x), app(app(rev2, y), l)))
APP(app(rev2, x), app(app(cons, y), l)) → APP(app(rev2, y), l)
app(app(rev2, x), nil) → nil
app(app(rev2, x), app(app(cons, y), l)) → app(rev, app(app(cons, x), app(app(rev2, y), l)))
app(rev, app(app(cons, x), l)) → app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) → 0
app(app(rev1, app(s, x)), nil) → app(s, x)
app(app(rev1, x), app(app(cons, y), l)) → app(app(rev1, y), l)
app(rev, nil)
app(rev, app(app(cons, x0), x1))
app(app(rev1, 0), nil)
app(app(rev1, app(s, x0)), nil)
app(app(rev1, x0), app(app(cons, x1), x2))
app(app(rev2, x0), nil)
app(app(rev2, x0), app(app(cons, x1), x2))
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)
rev1(cons(x, l)) → rev21(x, l)
rev21(x, cons(y, l)) → rev1(cons(x, rev2(y, l)))
rev21(x, cons(y, l)) → rev21(y, l)
rev2(x, nil) → nil
rev2(x, cons(y, l)) → rev(cons(x, rev2(y, l)))
rev(cons(x, l)) → cons(rev1(x, l), rev2(x, l))
rev1(0, nil) → 0
rev1(s(x), nil) → s(x)
rev1(x, cons(y, l)) → rev1(y, l)
rev(nil)
rev(cons(x0, x1))
rev1(0, nil)
rev1(s(x0), nil)
rev1(x0, cons(x1, x2))
rev2(x0, nil)
rev2(x0, cons(x1, x2))
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)
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)
rev1(cons(x, l)) → rev21(x, l)
rev21(x, cons(y, l)) → rev1(cons(x, rev2(y, l)))
rev21(x, cons(y, l)) → rev21(y, l)
rev2(x, nil) → nil
rev2(x, cons(y, l)) → rev(cons(x, rev2(y, l)))
rev(cons(x, l)) → cons(rev1(x, l), rev2(x, l))
rev1(0, nil) → 0
rev1(s(x), nil) → s(x)
rev1(x, cons(y, l)) → rev1(y, l)
rev(nil)
rev(cons(x0, x1))
rev1(0, nil)
rev1(s(x0), nil)
rev1(x0, cons(x1, x2))
rev2(x0, nil)
rev2(x0, cons(x1, x2))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
rev21(x, cons(y, l)) → rev1(cons(x, rev2(y, l)))
rev21(x, cons(y, l)) → rev21(y, l)
POL(0) = 1
POL(cons(x1, x2)) = 1 + x2
POL(nil) = 1
POL(rev(x1)) = x1
POL(rev1(x1)) = x1
POL(rev1(x1, x2)) = 1 + x2
POL(rev2(x1, x2)) = x2
POL(rev21(x1, x2)) = 1 + x2
POL(s(x1)) = 1
rev2(x, cons(y, l)) → rev(cons(x, rev2(y, l)))
rev(cons(x, l)) → cons(rev1(x, l), rev2(x, l))
rev2(x, nil) → nil
rev1(cons(x, l)) → rev21(x, l)
rev2(x, nil) → nil
rev2(x, cons(y, l)) → rev(cons(x, rev2(y, l)))
rev(cons(x, l)) → cons(rev1(x, l), rev2(x, l))
rev1(0, nil) → 0
rev1(s(x), nil) → s(x)
rev1(x, cons(y, l)) → rev1(y, l)
rev(nil)
rev(cons(x0, x1))
rev1(0, nil)
rev1(s(x0), nil)
rev1(x0, cons(x1, x2))
rev2(x0, nil)
rev2(x0, cons(x1, x2))
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
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(rev, nil) → nil
app(rev, app(app(cons, x), l)) → app(app(cons, app(app(rev1, x), l)), app(app(rev2, x), l))
app(app(rev1, 0), nil) → 0
app(app(rev1, app(s, x)), nil) → app(s, x)
app(app(rev1, x), app(app(cons, y), l)) → app(app(rev1, y), l)
app(app(rev2, x), nil) → nil
app(app(rev2, x), app(app(cons, y), l)) → app(rev, app(app(cons, x), app(app(rev2, y), l)))
app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
app(rev, nil)
app(rev, app(app(cons, x0), x1))
app(app(rev1, 0), nil)
app(app(rev1, app(s, x0)), nil)
app(app(rev1, x0), app(app(cons, x1), x2))
app(app(rev2, x0), nil)
app(app(rev2, x0), app(app(cons, x1), x2))
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(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
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(rev, nil)
app(rev, app(app(cons, x0), x1))
app(app(rev1, 0), nil)
app(app(rev1, app(s, x0)), nil)
app(app(rev1, x0), app(app(cons, x1), x2))
app(app(rev2, x0), nil)
app(app(rev2, x0), app(app(cons, x1), x2))
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)
From the DPs we obtained the following set of size-change graphs: