(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0) → cons(0, f(s(0)))
f(s(0)) → f(p(s(0)))
p(s(0)) → 0
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(0) → F(s(0))
F(s(0)) → F(p(s(0)))
F(s(0)) → P(s(0))
The TRS R consists of the following rules:
f(0) → cons(0, f(s(0)))
f(s(0)) → f(p(s(0)))
p(s(0)) → 0
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 1 less node.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(0)) → F(p(s(0)))
F(0) → F(s(0))
The TRS R consists of the following rules:
f(0) → cons(0, f(s(0)))
f(s(0)) → f(p(s(0)))
p(s(0)) → 0
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(0)[ ]n[ ] → F(0)[ ]n[ ]
This rule is correct for the QDP as the following derivation shows:
F(0)[ ]n[ ] → F(0)[ ]n[ ]
by Narrowing at position: []
F(0)[ ]n[ ] → F(s(0))[ ]n[ ]
by OriginalRule from TRS P
F(s(0))[ ]n[ ] → F(0)[ ]n[ ]
by Narrowing at position: [0]
F(s(0))[ ]n[ ] → F(p(s(0)))[ ]n[ ]
by OriginalRule from TRS P
p(s(0))[ ]n[ ] → 0[ ]n[ ]
by OriginalRule from TRS R
(6) NO