It is also worth noting that the work of this paper can be viewed as an interesting proof for the equivalence between (13) and (15), because both Theorem 1 and Proposition 2 represent the same eigenvalue distribution.
If ∑ i = 1 ∞ ( 1 - b i ) λ < ∞ and f: S → S is a strong (b n )-contraction, then there exists a unique fixed point x ∈ S of the mapping f and x = lim n → ∞ f n p for every p∈ S. Proof: From equivalence (3), we have ∑ i = 1 ∞ ( 1 - b i ) λ < ∞ ⇔ lim n → ∞ ( T λ D ) i = n ∞ b i = 1.
If ∑ i = 1 ∞ ( 1 - b i ) < ∞ and f: S → S is a strong (b n )-contraction, then there exists a unique fixed point x ∈ S of the mapping f and x = lim n → ∞ f n p for every p∈ S. Proof: From equivalence (4), we have ∑ i = 1 ∞ ( 1 - b i ) < ∞ ⇔ lim n → ∞ ( T λ S W ) i = n ∞ b i = 1.
If ∑ i = 1 ∞ ( 1 - b i ) λ < ∞ and f: S → S is a strong (b n )-contraction, then there exists a unique fixed point x ∈ S of the mapping f and x = lim n → ∞ f n p for every p∈ S. Proof: From equivalence (5), we have ∑ i = 1 ∞ ( 1 - b i ) λ < ∞ ⇔ lim n → ∞ ( T λ A A ) i = n ∞ b i = 1.
The logic of Equalities with Uninterpreted Functions is used in the formal verification community mainly for proofs of equivalence: proving that two versions of a hardware design are the same, or that input and output of a compiler are semantically equivalent are two prominent examples of such proofs.
The proof of equivalence was accomplished using the formulas simplified in this paper.
The proof of equivalence relies heavily on the character theory of finite Abelian groups.
For five analytes, lysine, phosphorus, potassium, vitamin B6 and vitamin E, equivalence was more likely than not, but a strict proof of equivalence cannot be given.
The proof of equivalence and proof of non-equivalence tests (tests E2 and E3, respectively) are seen to have the nominal size (0.05) at the equivalence/non-equivalence borderline value dif = 27.2%.
The absence of a detectable difference suggests equivalence, but proof of equivalence would require a different sample size.
