(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(tt, x) → f(isList(x), x)
isList(Cons(x, xs)) → isList(xs)
isList(nil) → tt
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:
F(tt, x) → F(isList(x), x)
F(tt, x) → ISLIST(x)
ISLIST(Cons(x, xs)) → ISLIST(xs)
The TRS R consists of the following rules:
f(tt, x) → f(isList(x), x)
isList(Cons(x, xs)) → isList(xs)
isList(nil) → tt
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:
F(tt, x) → F(isList(x), x)
The TRS R consists of the following rules:
f(tt, x) → f(isList(x), x)
isList(Cons(x, xs)) → isList(xs)
isList(nil) → tt
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' = [ ] on the rule
F(tt, nil)[ ]n[ ] → F(tt, nil)[ ]n[ ]
This rule is correct for the QDP as the following derivation shows:
F(tt, nil)[ ]n[ ] → F(tt, nil)[ ]n[ ]
by Narrowing at position: [0]
intermediate steps: Instantiation - Instantiation
F(tt, x)[ ]n[ ] → F(isList(x), x)[ ]n[ ]
by OriginalRule from TRS P
isList(nil)[ ]n[ ] → tt[ ]n[ ]
by OriginalRule from TRS R
(6) NO