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 QDPSizeChangeProof (⇔)
↳24 TRUE
↳25 QDP
↳26 UsableRulesProof (⇔)
↳27 QDP
↳28 QDPSizeChangeProof (⇔)
↳29 TRUE
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(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(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))
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(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(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))
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(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
APP(app(le, app(s, x)), app(s, y)) → APP(le, x)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(if, app(app(le, x), y))
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(le, x), y)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(le, x)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(maxlist, y), ys)
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(maxlist, y)
APP(height, app(app(node, x), xs)) → APP(s, app(app(maxlist, 0), app(app(map, height), xs)))
APP(height, app(app(node, x), xs)) → APP(app(maxlist, 0), app(app(map, height), xs))
APP(height, app(app(node, x), xs)) → APP(maxlist, 0)
APP(height, app(app(node, x), xs)) → APP(app(map, height), xs)
APP(height, app(app(node, x), xs)) → APP(map, height)
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(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(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
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(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(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))
APP(app(le, app(s, x)), app(s, y)) → APP(app(le, x), y)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))
le1(s(x), s(y)) → le1(x, y)
map(x0, nil)
map(x0, cons(x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
maxlist(x0, cons(x1, x2))
maxlist(x0, nil)
height(node(x0, x1))
map(x0, nil)
map(x0, cons(x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
maxlist(x0, cons(x1, x2))
maxlist(x0, nil)
height(node(x0, x1))
le1(s(x), s(y)) → le1(x, y)
From the DPs we obtained the following set of size-change graphs:
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(maxlist, y), ys)
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(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(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))
APP(app(maxlist, x), app(app(cons, y), ys)) → APP(app(maxlist, y), ys)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))
maxlist1(x, cons(y, ys)) → maxlist1(y, ys)
map(x0, nil)
map(x0, cons(x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
maxlist(x0, cons(x1, x2))
maxlist(x0, nil)
height(node(x0, x1))
map(x0, nil)
map(x0, cons(x1, x2))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
maxlist(x0, cons(x1, x2))
maxlist(x0, nil)
height(node(x0, x1))
maxlist1(x, cons(y, ys)) → maxlist1(y, ys)
From the DPs we obtained the following set of size-change graphs:
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(height, app(app(node, x), xs)) → APP(app(map, height), xs)
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(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(maxlist, x), app(app(cons, y), ys)) → app(app(if, app(app(le, x), y)), app(app(maxlist, y), ys))
app(app(maxlist, x), nil) → x
app(height, app(app(node, x), xs)) → app(s, app(app(maxlist, 0), app(app(map, height), xs)))
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))
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(height, app(app(node, x), xs)) → APP(app(map, height), xs)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(le, 0), x0)
app(app(le, app(s, x0)), 0)
app(app(le, app(s, x0)), app(s, x1))
app(app(maxlist, x0), app(app(cons, x1), x2))
app(app(maxlist, x0), nil)
app(height, app(app(node, x0), x1))
From the DPs we obtained the following set of size-change graphs: