(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
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(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) → x
app(app(div, 0), app(s, y)) → 0
app(app(div, app(s, x)), app(s, y)) → app(s, app(app(div, app(app(minus, x), app(id, y))), app(s, y)))
app(id, x) → x
app(id, x) → app(s, app(s, app(s, x)))
app(id, app(p, x)) → app(id, app(s, app(id, x)))
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
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(minus, app(s, x)), app(s, y)) → APP(app(minus, app(p, app(s, x))), app(p, app(s, y)))
APP(app(minus, app(s, x)), app(s, y)) → APP(minus, app(p, app(s, x)))
APP(app(minus, app(s, x)), app(s, y)) → APP(p, app(s, x))
APP(app(minus, app(s, x)), app(s, y)) → APP(p, app(s, y))
APP(app(div, app(s, x)), app(s, y)) → APP(s, app(app(div, app(app(minus, x), app(id, y))), app(s, y)))
APP(app(div, app(s, x)), app(s, y)) → APP(app(div, app(app(minus, x), app(id, y))), app(s, y))
APP(app(div, app(s, x)), app(s, y)) → APP(div, app(app(minus, x), app(id, y)))
APP(app(div, app(s, x)), app(s, y)) → APP(app(minus, x), app(id, y))
APP(app(div, app(s, x)), app(s, y)) → APP(minus, x)
APP(app(div, app(s, x)), app(s, y)) → APP(id, y)
APP(id, x) → APP(s, app(s, app(s, x)))
APP(id, x) → APP(s, app(s, x))
APP(id, x) → APP(s, x)
APP(id, app(p, x)) → APP(id, app(s, app(id, x)))
APP(id, app(p, x)) → APP(s, app(id, x))
APP(id, app(p, x)) → APP(id, x)
The TRS R consists of the following rules:
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(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) → x
app(app(div, 0), app(s, y)) → 0
app(app(div, app(s, x)), app(s, y)) → app(s, app(app(div, app(app(minus, x), app(id, y))), app(s, y)))
app(id, x) → x
app(id, x) → app(s, app(s, app(s, x)))
app(id, app(p, x)) → app(id, app(s, app(id, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 15 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(id, app(p, x)) → APP(id, x)
The TRS R consists of the following rules:
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(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) → x
app(app(div, 0), app(s, y)) → 0
app(app(div, app(s, x)), app(s, y)) → app(s, app(app(div, app(app(minus, x), app(id, y))), app(s, y)))
app(id, x) → x
app(id, x) → app(s, app(s, app(s, x)))
app(id, app(p, x)) → app(id, app(s, app(id, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(minus, app(s, x)), app(s, y)) → APP(app(minus, app(p, app(s, x))), app(p, app(s, y)))
The TRS R consists of the following rules:
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(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) → x
app(app(div, 0), app(s, y)) → 0
app(app(div, app(s, x)), app(s, y)) → app(s, app(app(div, app(app(minus, x), app(id, y))), app(s, y)))
app(id, x) → x
app(id, x) → app(s, app(s, app(s, x)))
app(id, app(p, x)) → app(id, app(s, app(id, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(div, app(s, x)), app(s, y)) → APP(app(div, app(app(minus, x), app(id, y))), app(s, y))
The TRS R consists of the following rules:
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(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) → x
app(app(div, 0), app(s, y)) → 0
app(app(div, app(s, x)), app(s, y)) → app(s, app(app(div, app(app(minus, x), app(id, y))), app(s, y)))
app(id, x) → x
app(id, x) → app(s, app(s, app(s, x)))
app(id, app(p, x)) → app(id, app(s, app(id, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
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)
The TRS R consists of the following rules:
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(minus, x), 0) → x
app(app(minus, app(s, x)), app(s, y)) → app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) → x
app(app(div, 0), app(s, y)) → 0
app(app(div, app(s, x)), app(s, y)) → app(s, app(app(div, app(app(minus, x), app(id, y))), app(s, y)))
app(id, x) → x
app(id, x) → app(s, app(s, app(s, x)))
app(id, app(p, x)) → app(id, app(s, app(id, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.