0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 QDPSizeChangeProof (⇔)
↳6 TRUE
app(id, x) → x
app(add, 0) → id
app(app(add, app(s, x)), y) → app(s, app(app(add, 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(add, app(s, x)), y) → APP(s, app(app(add, x), y))
APP(app(add, app(s, x)), y) → APP(app(add, x), y)
APP(app(add, app(s, x)), y) → APP(add, x)
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(id, x) → x
app(add, 0) → id
app(app(add, app(s, x)), y) → app(s, app(app(add, 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(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(add, app(s, x)), y) → APP(app(add, x), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
app(id, x) → x
app(add, 0) → id
app(app(add, app(s, x)), y) → app(s, app(app(add, 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))
Order:Combined order from the following AFS and order.
app(x1, x2) = app(x2)
add = add
0 = 0
id = id
map = map
cons = cons
s = s
Lexicographic path order with status [LPO].
Quasi-Precedence:
app1 > id
app1: [1]
add: []
0: []
id: []
map: []
cons: []
s: []
AFS:
app(x1, x2) = app(x2)
add = add
0 = 0
id = id
map = map
cons = cons
s = s
From the DPs we obtained the following set of size-change graphs:
We oriented the following set of usable rules [AAECC05,FROCOS05].
app(add, 0) → id