(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(append, nil), ys) → ys
app(app(append, app(app(cons, x), xs)), ys) → app(app(cons, x), app(app(append, xs), ys))
app(app(flatwith, f), app(leaf, x)) → app(app(cons, app(f, x)), nil)
app(app(flatwith, f), app(node, xs)) → app(app(flatwithsub, f), xs)
app(app(flatwithsub, f), nil) → nil
app(app(flatwithsub, f), app(app(cons, x), xs)) → app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs))
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(append, nil), ys) → ys
app(app(append, app(app(cons, x), xs)), ys) → app(app(cons, x), app(app(append, xs), ys))
app(app(flatwith, f), app(leaf, x)) → app(app(cons, app(f, x)), nil)
app(app(flatwith, f), app(node, xs)) → app(app(flatwithsub, f), xs)
app(app(flatwithsub, f), nil) → nil
app(app(flatwithsub, f), app(app(cons, x), xs)) → app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs))
The set Q consists of the following terms:
app(app(append, nil), x0)
app(app(append, app(app(cons, x0), x1)), x2)
app(app(flatwith, x0), app(leaf, x1))
app(app(flatwith, x0), app(node, x1))
app(app(flatwithsub, x0), nil)
app(app(flatwithsub, x0), app(app(cons, x1), x2))
(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(append, app(app(cons, x), xs)), ys) → APP(app(cons, x), app(app(append, xs), ys))
APP(app(append, app(app(cons, x), xs)), ys) → APP(app(append, xs), ys)
APP(app(append, app(app(cons, x), xs)), ys) → APP(append, xs)
APP(app(flatwith, f), app(leaf, x)) → APP(app(cons, app(f, x)), nil)
APP(app(flatwith, f), app(leaf, x)) → APP(cons, app(f, x))
APP(app(flatwith, f), app(leaf, x)) → APP(f, x)
APP(app(flatwith, f), app(node, xs)) → APP(app(flatwithsub, f), xs)
APP(app(flatwith, f), app(node, xs)) → APP(flatwithsub, f)
APP(app(flatwithsub, f), app(app(cons, x), xs)) → APP(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs))
APP(app(flatwithsub, f), app(app(cons, x), xs)) → APP(append, app(app(flatwith, f), x))
APP(app(flatwithsub, f), app(app(cons, x), xs)) → APP(app(flatwith, f), x)
APP(app(flatwithsub, f), app(app(cons, x), xs)) → APP(flatwith, f)
APP(app(flatwithsub, f), app(app(cons, x), xs)) → APP(app(flatwithsub, f), xs)
The TRS R consists of the following rules:
app(app(append, nil), ys) → ys
app(app(append, app(app(cons, x), xs)), ys) → app(app(cons, x), app(app(append, xs), ys))
app(app(flatwith, f), app(leaf, x)) → app(app(cons, app(f, x)), nil)
app(app(flatwith, f), app(node, xs)) → app(app(flatwithsub, f), xs)
app(app(flatwithsub, f), nil) → nil
app(app(flatwithsub, f), app(app(cons, x), xs)) → app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs))
The set Q consists of the following terms:
app(app(append, nil), x0)
app(app(append, app(app(cons, x0), x1)), x2)
app(app(flatwith, x0), app(leaf, x1))
app(app(flatwith, x0), app(node, x1))
app(app(flatwithsub, x0), nil)
app(app(flatwithsub, x0), app(app(cons, x1), x2))
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 8 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(append, app(app(cons, x), xs)), ys) → APP(app(append, xs), ys)
The TRS R consists of the following rules:
app(app(append, nil), ys) → ys
app(app(append, app(app(cons, x), xs)), ys) → app(app(cons, x), app(app(append, xs), ys))
app(app(flatwith, f), app(leaf, x)) → app(app(cons, app(f, x)), nil)
app(app(flatwith, f), app(node, xs)) → app(app(flatwithsub, f), xs)
app(app(flatwithsub, f), nil) → nil
app(app(flatwithsub, f), app(app(cons, x), xs)) → app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs))
The set Q consists of the following terms:
app(app(append, nil), x0)
app(app(append, app(app(cons, x0), x1)), x2)
app(app(flatwith, x0), app(leaf, x1))
app(app(flatwith, x0), app(node, x1))
app(app(flatwithsub, x0), nil)
app(app(flatwithsub, x0), app(app(cons, x1), x2))
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(flatwith, f), app(node, xs)) → APP(app(flatwithsub, f), xs)
APP(app(flatwithsub, f), app(app(cons, x), xs)) → APP(app(flatwith, f), x)
APP(app(flatwith, f), app(leaf, x)) → APP(f, x)
APP(app(flatwithsub, f), app(app(cons, x), xs)) → APP(app(flatwithsub, f), xs)
The TRS R consists of the following rules:
app(app(append, nil), ys) → ys
app(app(append, app(app(cons, x), xs)), ys) → app(app(cons, x), app(app(append, xs), ys))
app(app(flatwith, f), app(leaf, x)) → app(app(cons, app(f, x)), nil)
app(app(flatwith, f), app(node, xs)) → app(app(flatwithsub, f), xs)
app(app(flatwithsub, f), nil) → nil
app(app(flatwithsub, f), app(app(cons, x), xs)) → app(app(append, app(app(flatwith, f), x)), app(app(flatwithsub, f), xs))
The set Q consists of the following terms:
app(app(append, nil), x0)
app(app(append, app(app(cons, x0), x1)), x2)
app(app(flatwith, x0), app(leaf, x1))
app(app(flatwith, x0), app(node, x1))
app(app(flatwithsub, x0), nil)
app(app(flatwithsub, x0), app(app(cons, x1), x2))
We have to consider all minimal (P,Q,R)-chains.