(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(app(app(rec, t), u), v), 0) → t
app(app(app(app(rec, t), u), v), app(s, x)) → app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) → app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) → app(app(app(app(rec, t), u), v), n)
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(app(app(rec, t), u), v), 0) → t
app(app(app(app(rec, t), u), v), app(s, x)) → app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) → app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) → app(app(app(app(rec, t), u), v), n)
The set Q consists of the following terms:
app(app(app(app(rec, x0), x1), x2), 0)
app(app(app(app(rec, x0), x1), x2), app(s, x3))
app(app(app(app(rec, x0), x1), x2), app(lim, x3))
app(app(app(app(rectuv, x0), x1), x2), n)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(v, f)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(app(app(rectuv, t), u), v), app(f, n))
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(app(rectuv, t), u), v)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(rectuv, t), u)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(rectuv, t)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(f, n)
APP(app(app(app(rectuv, t), u), v), n) → APP(app(app(app(rec, t), u), v), n)
APP(app(app(app(rectuv, t), u), v), n) → APP(app(app(rec, t), u), v)
APP(app(app(app(rectuv, t), u), v), n) → APP(app(rec, t), u)
APP(app(app(app(rectuv, t), u), v), n) → APP(rec, t)
The TRS R consists of the following rules:
app(app(app(app(rec, t), u), v), 0) → t
app(app(app(app(rec, t), u), v), app(s, x)) → app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) → app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) → app(app(app(app(rec, t), u), v), n)
The set Q consists of the following terms:
app(app(app(app(rec, x0), x1), x2), 0)
app(app(app(app(rec, x0), x1), x2), app(s, x3))
app(app(app(app(rec, x0), x1), x2), app(lim, x3))
app(app(app(app(rectuv, x0), x1), x2), n)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(u, x)
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(app(u, x), app(app(app(app(rec, t), u), v), x))
APP(app(app(app(rec, t), u), v), app(s, x)) → APP(app(app(app(rec, t), u), v), x)
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
APP(app(app(app(rec, t), u), v), app(lim, f)) → APP(v, f)
The TRS R consists of the following rules:
app(app(app(app(rec, t), u), v), 0) → t
app(app(app(app(rec, t), u), v), app(s, x)) → app(app(u, x), app(app(app(app(rec, t), u), v), x))
app(app(app(app(rec, t), u), v), app(lim, f)) → app(app(v, f), app(app(app(app(rectuv, t), u), v), app(f, n)))
app(app(app(app(rectuv, t), u), v), n) → app(app(app(app(rec, t), u), v), n)
The set Q consists of the following terms:
app(app(app(app(rec, x0), x1), x2), 0)
app(app(app(app(rec, x0), x1), x2), app(s, x3))
app(app(app(app(rec, x0), x1), x2), app(lim, x3))
app(app(app(app(rectuv, x0), x1), x2), n)
We have to consider all minimal (P,Q,R)-chains.