(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
app(app(iterate, f), x) → app(app(cons, x), app(app(iterate, f), app(f, x)))
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(iterate, f), x) → APP(app(cons, x), app(app(iterate, f), app(f, x)))
APP(app(iterate, f), x) → APP(cons, x)
APP(app(iterate, f), x) → APP(app(iterate, f), app(f, x))
APP(app(iterate, f), x) → APP(f, x)
The TRS R consists of the following rules:
app(app(iterate, f), x) → app(app(cons, x), app(app(iterate, f), app(f, x)))
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 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP(app(iterate, f), x) → APP(f, x)
APP(app(iterate, f), x) → APP(app(iterate, f), app(f, x))
The TRS R consists of the following rules:
app(app(iterate, f), x) → app(app(cons, x), app(app(iterate, f), app(f, x)))
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 / app(x0, x1)] on the rule
APP(app(iterate, x0), app(x0, x1))[ ]n[ ] → APP(app(iterate, x0), app(x0, x1))[ ]n[x1 / app(x0, x1)]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu - Instantiation
APP(app(iterate, f), x)[ ]n[ ] → APP(app(iterate, f), app(f, x))[ ]n[ ]
by OriginalRule from TRS P
(6) NO