0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
↳5 QDP
↳6 UsableRulesProof (⇔)
↳7 QDP
↳8 ATransformationProof (⇔)
↳9 QDP
↳10 QDPSizeChangeProof (⇔)
↳11 TRUE
↳12 QDP
↳13 UsableRulesProof (⇔)
↳14 QDP
↳15 ATransformationProof (⇔)
↳16 QDP
↳17 QDPSizeChangeProof (⇔)
↳18 TRUE
↳19 QDP
↳20 QDPApplicativeOrderProof (⇔)
↳21 QDP
↳22 QDPApplicativeOrderProof (⇔)
↳23 QDP
↳24 PisEmptyProof (⇔)
↳25 TRUE
↳26 QDP
↳27 QDPSizeChangeProof (⇔)
↳28 TRUE
app(app(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, x), y)
app(double, 0) → 0
app(double, app(s, x)) → app(s, app(s, app(double, x)))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) → app(app(plus, x), app(s, y))
app(app(plus, app(s, x)), y) → app(s, app(app(plus, app(app(minus, x), y)), app(double, 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(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(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(double, app(s, x)) → APP(s, app(s, app(double, x)))
APP(double, app(s, x)) → APP(s, app(double, x))
APP(double, app(s, x)) → APP(double, x)
APP(app(plus, app(s, x)), y) → APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) → APP(plus, x)
APP(app(plus, app(s, x)), y) → APP(app(plus, x), app(s, y))
APP(app(plus, app(s, x)), y) → APP(s, y)
APP(app(plus, app(s, x)), y) → APP(s, app(app(plus, app(app(minus, x), y)), app(double, y)))
APP(app(plus, app(s, x)), y) → APP(app(plus, app(app(minus, x), y)), app(double, y))
APP(app(plus, app(s, x)), y) → APP(plus, app(app(minus, x), y))
APP(app(plus, app(s, x)), y) → APP(app(minus, x), y)
APP(app(plus, app(s, x)), y) → APP(minus, x)
APP(app(plus, app(s, x)), y) → APP(double, 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(app(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, x), y)
app(double, 0) → 0
app(double, app(s, x)) → app(s, app(s, app(double, x)))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) → app(app(plus, x), app(s, y))
app(app(plus, app(s, x)), y) → app(s, app(app(plus, app(app(minus, x), y)), app(double, 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(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(double, app(s, x)) → APP(double, x)
app(app(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, x), y)
app(double, 0) → 0
app(double, app(s, x)) → app(s, app(s, app(double, x)))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) → app(app(plus, x), app(s, y))
app(app(plus, app(s, x)), y) → app(s, app(app(plus, app(app(minus, x), y)), app(double, 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(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(double, app(s, x)) → APP(double, x)
double(s(x)) → double(x)
From the DPs we obtained the following set of size-change graphs:
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(double, 0) → 0
app(double, app(s, x)) → app(s, app(s, app(double, x)))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) → app(app(plus, x), app(s, y))
app(app(plus, app(s, x)), y) → app(s, app(app(plus, app(app(minus, x), y)), app(double, 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(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(app(minus, app(s, x)), app(s, y)) → APP(app(minus, x), y)
minus(s(x), s(y)) → minus(x, y)
From the DPs we obtained the following set of size-change graphs:
APP(app(plus, app(s, x)), y) → APP(app(plus, x), app(s, y))
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) → APP(app(plus, app(app(minus, x), y)), app(double, y))
app(app(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, x), y)
app(double, 0) → 0
app(double, app(s, x)) → app(s, app(s, app(double, x)))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) → app(app(plus, x), app(s, y))
app(app(plus, app(s, x)), y) → app(s, app(app(plus, app(app(minus, x), y)), app(double, 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(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)
plus1(s(x), y) → plus1(x, s(y))
plus1(s(x), y) → plus1(x, y)
plus1(s(x), y) → plus1(minus(x, y), double(y))
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(plus, app(s, x)), y) → APP(app(plus, x), app(s, y))
APP(app(plus, app(s, x)), y) → APP(app(plus, x), y)
Used ordering: Polynomial interpretation [POLO]:
APP(app(plus, app(s, x)), y) → APP(app(plus, app(app(minus, x), y)), app(double, y))
POL(0) = 0
POL(double(x1)) = 0
POL(minus(x1, x2)) = 1 + x1
POL(plus1(x1, x2)) = x1
POL(s(x1)) = 1 + x1
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, x), y)
app(app(minus, x), 0) → x
APP(app(plus, app(s, x)), y) → APP(app(plus, app(app(minus, x), y)), app(double, y))
app(app(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, x), y)
app(double, 0) → 0
app(double, app(s, x)) → app(s, app(s, app(double, x)))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) → app(app(plus, x), app(s, y))
app(app(plus, app(s, x)), y) → app(s, app(app(plus, app(app(minus, x), y)), app(double, 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(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)
plus1(s(x), y) → plus1(minus(x, y), double(y))
double(0) → 0
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(plus, app(s, x)), y) → APP(app(plus, app(app(minus, x), y)), app(double, y))
POL(0) = 0
POL(double(x1)) = x1
POL(minus(x1, x2)) = x1
POL(plus1(x1, x2)) = x1
POL(s(x1)) = 1 + x1
app(double, 0) → 0
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, x), y)
app(app(minus, x), 0) → x
app(app(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, x), y)
app(double, 0) → 0
app(double, app(s, x)) → app(s, app(s, app(double, x)))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) → app(app(plus, x), app(s, y))
app(app(plus, app(s, x)), y) → app(s, app(app(plus, app(app(minus, x), y)), app(double, 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(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(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(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(cons, 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(double, 0) → 0
app(double, app(s, x)) → app(s, app(s, app(double, x)))
app(app(plus, 0), y) → y
app(app(plus, app(s, x)), y) → app(s, app(app(plus, x), y))
app(app(plus, app(s, x)), y) → app(app(plus, x), app(s, y))
app(app(plus, app(s, x)), y) → app(s, app(app(plus, app(app(minus, x), y)), app(double, 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(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)
From the DPs we obtained the following set of size-change graphs: