(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(f, 0), n) → app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))
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(f, 0), n) → APP(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
APP(app(f, 0), n) → APP(hd, app(app(map, f), app(app(cons, 0), nil)))
APP(app(f, 0), n) → APP(app(map, f), app(app(cons, 0), nil))
APP(app(f, 0), n) → APP(map, f)
APP(app(f, 0), n) → APP(app(cons, 0), nil)
APP(app(f, 0), n) → APP(cons, 0)
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)
The TRS R consists of the following rules:
app(app(f, 0), n) → app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))
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 1 SCC with 4 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(f, 0), n) → APP(app(map, f), app(app(cons, 0), nil))
APP(app(f, 0), n) → APP(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
APP(app(f, 0), n) → APP(app(cons, 0), nil)
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(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
The TRS R consists of the following rules:
app(app(f, 0), n) → app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) NonLoopProof (EQUIVALENT transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0,
σ' = [ ], and μ' = [x1 / cons, x0 / nil] on the rule
APP(app(cons, 0), nil)[ ]n[ ] → APP(app(cons, 0), nil)[ ]n[x1 / cons, x0 / nil]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu - Instantiation
APP(app(f, 0), n)[ ]n[ ] → APP(app(cons, 0), nil)[ ]n[ ]
by OriginalRule from TRS P
(6) NO