(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(treemap, f), app(app(node, x), xs)) → app(app(node, app(f, x)), app(app(map, app(treemap, 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(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(treemap, f), app(app(node, x), xs)) → app(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))
The set Q consists of the following terms:
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(treemap, x0), app(app(node, 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(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(treemap, f), app(app(node, x), xs)) → APP(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))
APP(app(treemap, f), app(app(node, x), xs)) → APP(node, app(f, x))
APP(app(treemap, f), app(app(node, x), xs)) → APP(f, x)
APP(app(treemap, f), app(app(node, x), xs)) → APP(app(map, app(treemap, f)), xs)
APP(app(treemap, f), app(app(node, x), xs)) → APP(map, app(treemap, f))
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(treemap, f), app(app(node, x), xs)) → app(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))
The set Q consists of the following terms:
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(treemap, x0), app(app(node, 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 1 SCC with 5 less nodes.
(6) 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)
APP(app(treemap, f), app(app(node, x), xs)) → APP(f, x)
APP(app(treemap, f), app(app(node, x), xs)) → APP(app(map, app(treemap, f)), xs)
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(treemap, f), app(app(node, x), xs)) → app(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))
The set Q consists of the following terms:
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(treemap, x0), app(app(node, x1), x2))
We have to consider all minimal (P,Q,R)-chains.