(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))
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(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))
The set Q consists of the following terms:
app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init
(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(compose, f), g), x) → APP(g, app(f, x))
APP(app(app(compose, f), g), x) → APP(f, x)
APP(reverse, l) → APP(app(reverse2, l), nil)
APP(reverse, l) → APP(reverse2, l)
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(app(reverse2, xs), app(app(cons, x), l))
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(reverse2, xs)
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(app(cons, x), l)
LAST → APP(app(compose, hd), reverse)
LAST → APP(compose, hd)
INIT → APP(app(compose, reverse), app(app(compose, tl), reverse))
INIT → APP(compose, reverse)
INIT → APP(app(compose, tl), reverse)
INIT → APP(compose, tl)
The TRS R consists of the following rules:
app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))
The set Q consists of the following terms:
app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 10 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(reverse2, app(app(cons, x), xs)), l) → APP(app(reverse2, xs), app(app(cons, x), l))
The TRS R consists of the following rules:
app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))
The set Q consists of the following terms:
app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init
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(app(compose, f), g), x) → APP(f, x)
APP(app(app(compose, f), g), x) → APP(g, app(f, x))
The TRS R consists of the following rules:
app(app(app(compose, f), g), x) → app(g, app(f, x))
app(reverse, l) → app(app(reverse2, l), nil)
app(app(reverse2, nil), l) → l
app(app(reverse2, app(app(cons, x), xs)), l) → app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) → x
app(tl, app(app(cons, x), xs)) → xs
last → app(app(compose, hd), reverse)
init → app(app(compose, reverse), app(app(compose, tl), reverse))
The set Q consists of the following terms:
app(app(app(compose, x0), x1), x2)
app(reverse, x0)
app(app(reverse2, nil), x0)
app(app(reverse2, app(app(cons, x0), x1)), x2)
app(hd, app(app(cons, x0), x1))
app(tl, app(app(cons, x0), x1))
last
init
We have to consider all minimal (P,Q,R)-chains.