(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, zeros)
tail(cons(X, XS)) → XS
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(cons(x1, x2)) = 2·x1 + 2·x2
POL(tail(x1)) = 1 + x1
POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
tail(cons(X, XS)) → XS
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, zeros)
Q is empty.
(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:
ZEROS → ZEROS
The TRS R consists of the following rules:
zeros → cons(0, zeros)
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
ZEROS[ ]n[ ] → ZEROS[ ]n[ ]
This rule is correct for the QDP as the following derivation shows:
ZEROS[ ]n[ ] → ZEROS[ ]n[ ]
by OriginalRule from TRS P
(6) NO