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 ATransformationProof (⇔)
↳29 QDP
↳30 QReductionProof (⇔)
↳31 QDP
↳32 QDPSizeChangeProof (⇔)
↳33 TRUE
↳34 QDP
↳35 UsableRulesProof (⇔)
↳36 QDP
↳37 ATransformationProof (⇔)
↳38 QDP
↳39 QReductionProof (⇔)
↳40 QDP
↳41 QDPSizeChangeProof (⇔)
↳42 TRUE
↳43 QDP
↳44 UsableRulesProof (⇔)
↳45 QDP
↳46 ATransformationProof (⇔)
↳47 QDP
↳48 QReductionProof (⇔)
↳49 QDP
↳50 QDPOrderProof (⇔)
↳51 QDP
↳52 PisEmptyProof (⇔)
↳53 TRUE
↳54 QDP
↳55 UsableRulesProof (⇔)
↳56 QDP
↳57 ATransformationProof (⇔)
↳58 QDP
↳59 QReductionProof (⇔)
↳60 QDP
↳61 QDPSizeChangeProof (⇔)
↳62 TRUE
↳63 QDP
↳64 UsableRulesProof (⇔)
↳65 QDP
↳66 ATransformationProof (⇔)
↳67 QDP
↳68 QReductionProof (⇔)
↳69 QDP
↳70 QDPOrderProof (⇔)
↳71 QDP
↳72 PisEmptyProof (⇔)
↳73 TRUE
↳74 QDP
↳75 QDPSizeChangeProof (⇔)
↳76 TRUE
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, 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, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, 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, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
APP'(app'(minus, app'(s, x)), app'(s, y)) → APP'(app'(minus, x), y)
APP'(app'(minus, app'(s, x)), app'(s, y)) → APP'(minus, x)
APP'(app'(quot, app'(s, x)), app'(s, y)) → APP'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, y)))
APP'(app'(quot, app'(s, x)), app'(s, y)) → APP'(app'(quot, app'(app'(minus, x), y)), app'(s, y))
APP'(app'(quot, app'(s, x)), app'(s, y)) → APP'(quot, app'(app'(minus, x), y))
APP'(app'(quot, app'(s, x)), app'(s, y)) → APP'(app'(minus, x), y)
APP'(app'(quot, app'(s, x)), app'(s, y)) → APP'(minus, x)
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'(app, app'(app'(add, n), x)), y) → APP'(app'(add, n), app'(app'(app, x), y))
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app'(app, x), y)
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app, x)
APP'(app'(low, n), app'(app'(add, m), x)) → APP'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
APP'(app'(low, n), app'(app'(add, m), x)) → APP'(app'(if_low, app'(app'(le, m), n)), n)
APP'(app'(low, n), app'(app'(add, m), x)) → APP'(if_low, app'(app'(le, m), n))
APP'(app'(low, n), app'(app'(add, m), x)) → APP'(app'(le, m), n)
APP'(app'(low, n), app'(app'(add, m), x)) → APP'(le, m)
APP'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → APP'(app'(add, m), app'(app'(low, n), x))
APP'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → APP'(app'(low, n), x)
APP'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → APP'(low, n)
APP'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → APP'(app'(low, n), x)
APP'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → APP'(low, n)
APP'(app'(high, n), app'(app'(add, m), x)) → APP'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
APP'(app'(high, n), app'(app'(add, m), x)) → APP'(app'(if_high, app'(app'(le, m), n)), n)
APP'(app'(high, n), app'(app'(add, m), x)) → APP'(if_high, app'(app'(le, m), n))
APP'(app'(high, n), app'(app'(add, m), x)) → APP'(app'(le, m), n)
APP'(app'(high, n), app'(app'(add, m), x)) → APP'(le, m)
APP'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → APP'(app'(high, n), x)
APP'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → APP'(high, n)
APP'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → APP'(app'(add, m), app'(app'(high, n), x))
APP'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → APP'(app'(high, n), x)
APP'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → APP'(high, n)
APP'(quicksort, app'(app'(add, n), x)) → APP'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
APP'(quicksort, app'(app'(add, n), x)) → APP'(app, app'(quicksort, app'(app'(low, n), x)))
APP'(quicksort, app'(app'(add, n), x)) → APP'(quicksort, app'(app'(low, n), x))
APP'(quicksort, app'(app'(add, n), x)) → APP'(app'(low, n), x)
APP'(quicksort, app'(app'(add, n), x)) → APP'(low, n)
APP'(quicksort, app'(app'(add, n), x)) → APP'(app'(add, n), app'(quicksort, app'(app'(high, n), x)))
APP'(quicksort, app'(app'(add, n), x)) → APP'(quicksort, app'(app'(high, n), x))
APP'(quicksort, app'(app'(add, n), x)) → APP'(app'(high, n), x)
APP'(quicksort, app'(app'(add, n), x)) → APP'(high, n)
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(app'(add, app'(f, x)), app'(app'(map, f), xs))
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(add, app'(f, x))
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(f, x)
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(app'(map, f), xs)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(app'(filter2, app'(f, x)), f), x)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(filter2, app'(f, x)), f)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(filter2, app'(f, x))
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(f, x)
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(app'(add, x), app'(app'(filter, f), xs))
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(add, 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'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, 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, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app'(app, x), y)
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, 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, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
APP'(app'(app, app'(app'(add, n), x)), y) → APP'(app'(app, x), y)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
app1(add(n, x), y) → app1(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
app1(add(n, x), y) → app1(x, y)
From the DPs we obtained the following set of size-change graphs:
APP'(app'(le, app'(s, x)), app'(s, y)) → APP'(app'(le, x), y)
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, 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, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
APP'(app'(le, app'(s, x)), app'(s, y)) → APP'(app'(le, x), y)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
le1(s(x), s(y)) → le1(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
le1(s(x), s(y)) → le1(x, y)
From the DPs we obtained the following set of size-change graphs:
APP'(app'(high, n), app'(app'(add, m), x)) → APP'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
APP'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → APP'(app'(high, n), x)
APP'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → APP'(app'(high, n), x)
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, 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, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
APP'(app'(high, n), app'(app'(add, m), x)) → APP'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
APP'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → APP'(app'(high, n), x)
APP'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → APP'(app'(high, n), x)
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'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
high1(n, add(m, x)) → if_high1(le(m, n), n, add(m, x))
if_high1(true, n, add(m, x)) → high1(n, x)
if_high1(false, n, add(m, x)) → high1(n, x)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
high1(n, add(m, x)) → if_high1(le(m, n), n, add(m, x))
if_high1(true, n, add(m, x)) → high1(n, x)
if_high1(false, n, add(m, x)) → high1(n, x)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
From the DPs we obtained the following set of size-change graphs:
APP'(app'(low, n), app'(app'(add, m), x)) → APP'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
APP'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → APP'(app'(low, n), x)
APP'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → APP'(app'(low, n), x)
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, 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, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
APP'(app'(low, n), app'(app'(add, m), x)) → APP'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
APP'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → APP'(app'(low, n), x)
APP'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → APP'(app'(low, n), x)
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'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
low1(n, add(m, x)) → if_low1(le(m, n), n, add(m, x))
if_low1(true, n, add(m, x)) → low1(n, x)
if_low1(false, n, add(m, x)) → low1(n, x)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
low1(n, add(m, x)) → if_low1(le(m, n), n, add(m, x))
if_low1(true, n, add(m, x)) → low1(n, x)
if_low1(false, n, add(m, x)) → low1(n, x)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
From the DPs we obtained the following set of size-change graphs:
APP'(quicksort, app'(app'(add, n), x)) → APP'(quicksort, app'(app'(high, n), x))
APP'(quicksort, app'(app'(add, n), x)) → APP'(quicksort, app'(app'(low, n), x))
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, 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, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
APP'(quicksort, app'(app'(add, n), x)) → APP'(quicksort, app'(app'(high, n), x))
APP'(quicksort, app'(app'(add, n), x)) → APP'(quicksort, app'(app'(low, n), x))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
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_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
quicksort1(add(n, x)) → quicksort1(high(n, x))
quicksort1(add(n, x)) → quicksort1(low(n, x))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(false, n, add(m, x)) → low(n, x)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
if_low(true, n, add(m, x)) → add(m, low(n, x))
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
quicksort1(add(n, x)) → quicksort1(high(n, x))
quicksort1(add(n, x)) → quicksort1(low(n, x))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(false, n, add(m, x)) → low(n, x)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
if_low(true, n, add(m, x)) → add(m, low(n, x))
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
quicksort1(add(n, x)) → quicksort1(high(n, x))
quicksort1(add(n, x)) → quicksort1(low(n, x))
POL(0) = 0
POL(add(x1, x2)) = 1 + x2
POL(false) = 0
POL(high(x1, x2)) = x2
POL(if_high(x1, x2, x3)) = x3
POL(if_low(x1, x2, x3)) = x3
POL(le(x1, x2)) = 0
POL(low(x1, x2)) = x2
POL(nil) = 0
POL(quicksort1(x1)) = x1
POL(s(x1)) = 0
POL(true) = 0
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
high(n, nil) → nil
if_low(true, n, add(m, x)) → add(m, low(n, x))
if_high(false, n, add(m, x)) → add(m, high(n, x))
low(n, nil) → nil
low(n, add(m, x)) → if_low(le(m, n), n, add(m, x))
if_low(false, n, add(m, x)) → low(n, x)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
if_low(true, n, add(m, x)) → add(m, low(n, x))
high(n, nil) → nil
high(n, add(m, x)) → if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) → high(n, x)
if_high(false, n, add(m, x)) → add(m, high(n, x))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
APP'(app'(minus, app'(s, x)), app'(s, y)) → APP'(app'(minus, x), y)
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, 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, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
APP'(app'(minus, app'(s, x)), app'(s, y)) → APP'(app'(minus, x), y)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
minus1(s(x), s(y)) → minus1(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
minus1(s(x), s(y)) → minus1(x, y)
From the DPs we obtained the following set of size-change graphs:
APP'(app'(quot, app'(s, x)), app'(s, y)) → APP'(app'(quot, app'(app'(minus, x), y)), app'(s, y))
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, 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, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
APP'(app'(quot, app'(s, x)), app'(s, y)) → APP'(app'(quot, app'(app'(minus, x), y)), app'(s, y))
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, x1), x2))
app'(app'(app'(app'(filter2, true), x0), x1), x2)
app'(app'(app'(app'(filter2, false), x0), x1), x2)
quot1(s(x), s(y)) → quot1(minus(x, y), s(y))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
quot(0, s(x0))
quot(s(x0), s(x1))
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
app(nil, x0)
app(add(x0, x1), x2)
low(x0, nil)
low(x0, add(x1, x2))
if_low(true, x0, add(x1, x2))
if_low(false, x0, add(x1, x2))
high(x0, nil)
high(x0, add(x1, x2))
if_high(true, x0, add(x1, x2))
if_high(false, x0, add(x1, x2))
quicksort(nil)
quicksort(add(x0, x1))
map(x0, nil)
map(x0, add(x1, x2))
filter(x0, nil)
filter(x0, add(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)
quot1(s(x), s(y)) → quot1(minus(x, y), s(y))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
quot1(s(x), s(y)) → quot1(minus(x, y), s(y))
POL(0) = 0
POL(minus(x1, x2)) = x1
POL(quot1(x1, x2)) = x1
POL(s(x1)) = 1 + x1
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
minus(x0, 0)
minus(s(x0), s(x1))
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(app'(map, f), xs)
APP'(app'(map, f), app'(app'(add, x), xs)) → APP'(f, x)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
APP'(app'(app'(app'(filter2, true), f), x), xs) → APP'(app'(filter, f), xs)
APP'(app'(filter, f), app'(app'(add, x), xs)) → APP'(f, x)
APP'(app'(app'(app'(filter2, false), f), x), xs) → APP'(app'(filter, f), xs)
app'(app'(minus, x), 0) → x
app'(app'(minus, app'(s, x)), app'(s, y)) → app'(app'(minus, x), y)
app'(app'(quot, 0), app'(s, y)) → 0
app'(app'(quot, app'(s, x)), app'(s, y)) → app'(s, app'(app'(quot, app'(app'(minus, x), y)), app'(s, 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, nil), y) → y
app'(app'(app, app'(app'(add, n), x)), y) → app'(app'(add, n), app'(app'(app, x), y))
app'(app'(low, n), nil) → nil
app'(app'(low, n), app'(app'(add, m), x)) → app'(app'(app'(if_low, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_low, true), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(low, n), x))
app'(app'(app'(if_low, false), n), app'(app'(add, m), x)) → app'(app'(low, n), x)
app'(app'(high, n), nil) → nil
app'(app'(high, n), app'(app'(add, m), x)) → app'(app'(app'(if_high, app'(app'(le, m), n)), n), app'(app'(add, m), x))
app'(app'(app'(if_high, true), n), app'(app'(add, m), x)) → app'(app'(high, n), x)
app'(app'(app'(if_high, false), n), app'(app'(add, m), x)) → app'(app'(add, m), app'(app'(high, n), x))
app'(quicksort, nil) → nil
app'(quicksort, app'(app'(add, n), x)) → app'(app'(app, app'(quicksort, app'(app'(low, n), x))), app'(app'(add, n), app'(quicksort, app'(app'(high, n), x))))
app'(app'(map, f), nil) → nil
app'(app'(map, f), app'(app'(add, x), xs)) → app'(app'(add, app'(f, x)), app'(app'(map, f), xs))
app'(app'(filter, f), nil) → nil
app'(app'(filter, f), app'(app'(add, x), xs)) → app'(app'(app'(app'(filter2, app'(f, x)), f), x), xs)
app'(app'(app'(app'(filter2, true), f), x), xs) → app'(app'(add, x), app'(app'(filter, f), xs))
app'(app'(app'(app'(filter2, false), f), x), xs) → app'(app'(filter, f), xs)
app'(app'(minus, x0), 0)
app'(app'(minus, app'(s, x0)), app'(s, x1))
app'(app'(quot, 0), app'(s, x0))
app'(app'(quot, 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, nil), x0)
app'(app'(app, app'(app'(add, x0), x1)), x2)
app'(app'(low, x0), nil)
app'(app'(low, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_low, false), x0), app'(app'(add, x1), x2))
app'(app'(high, x0), nil)
app'(app'(high, x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, true), x0), app'(app'(add, x1), x2))
app'(app'(app'(if_high, false), x0), app'(app'(add, x1), x2))
app'(quicksort, nil)
app'(quicksort, app'(app'(add, x0), x1))
app'(app'(map, x0), nil)
app'(app'(map, x0), app'(app'(add, x1), x2))
app'(app'(filter, x0), nil)
app'(app'(filter, x0), app'(app'(add, 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: