0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
↳5 QDP
↳6 QDPOrderProof (⇔)
↳7 QDP
↳8 PisEmptyProof (⇔)
↳9 TRUE
↳10 QDP
↳11 QDP
↳12 QDPOrderProof (⇔)
↳13 QDP
↳14 DependencyGraphProof (⇔)
↳15 TRUE
app(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
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(*, x), app(app(+, y), z)) → APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, x), app(app(+, y), z)) → APP(+, app(app(*, x), y))
APP(app(*, x), app(app(+, y), z)) → APP(app(*, x), y)
APP(app(*, x), app(app(+, y), z)) → APP(app(*, x), z)
APP(app(*, app(app(+, y), z)), x) → APP(app(+, app(app(*, x), y)), app(app(*, x), z))
APP(app(*, app(app(+, y), z)), x) → APP(+, app(app(*, x), y))
APP(app(*, app(app(+, y), z)), x) → APP(app(*, x), y)
APP(app(*, app(app(+, y), z)), x) → APP(*, x)
APP(app(*, app(app(+, y), z)), x) → APP(app(*, x), z)
APP(app(*, app(app(*, x), y)), z) → APP(app(*, x), app(app(*, y), z))
APP(app(*, app(app(*, x), y)), z) → APP(app(*, y), z)
APP(app(*, app(app(*, x), y)), z) → APP(*, y)
APP(app(+, app(app(+, x), y)), z) → APP(app(+, x), app(app(+, y), z))
APP(app(+, app(app(+, x), y)), z) → APP(app(+, y), z)
APP(app(+, app(app(+, x), y)), z) → APP(+, 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(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
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(+, app(app(+, x), y)), z) → APP(app(+, y), z)
APP(app(+, app(app(+, x), y)), z) → APP(app(+, x), app(app(+, y), z))
app(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
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)
+1(+(x, y), z) → +1(y, z)
+1(+(x, y), z) → +1(x, +(y, z))
+(+(x, y), z) → +(x, +(y, z))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(+, app(app(+, x), y)), z) → APP(app(+, y), z)
APP(app(+, app(app(+, x), y)), z) → APP(app(+, x), app(app(+, y), z))
+12 > +2
+2: [1,2]
+12: [1,2]
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
app(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
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(*, x), app(app(+, y), z)) → APP(app(*, x), z)
APP(app(*, x), app(app(+, y), z)) → APP(app(*, x), y)
APP(app(*, app(app(+, y), z)), x) → APP(app(*, x), y)
APP(app(*, app(app(+, y), z)), x) → APP(app(*, x), z)
APP(app(*, app(app(*, x), y)), z) → APP(app(*, x), app(app(*, y), z))
APP(app(*, app(app(*, x), y)), z) → APP(app(*, y), z)
app(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
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(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(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
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)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
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)
map > [APP1, app2, true, false]
filter > [APP1, app2, true, false]
filter2 > [APP1, app2, true, false]
APP1: [1]
cons: []
true: []
map: []
false: []
app2: [2,1]
filter2: []
filter: []
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(app(*, x), app(app(+, y), z)) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) → app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) → app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) → app(app(+, x), app(app(+, y), z))
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)