Sturm-Tarski Theorem
نویسنده
چکیده
We have formalized the Sturm-Tarski theorem (also referred as the Tarski theorem): ∑ x∈(a,b),P (x)=0 sign(Q(x)) = Var(SRemS(P, P ′Q; a, b)) where a < b are elements of R ∪ {−∞,∞} that are not roots of P , with P,Q ∈ R[x]. Note, the usual Sturm theorem is an instance of the Sturm-Tarski theorem with Q = 1. The proof is based on [1] and Cyril Cohen’s work in Coq [2]. With the Sturm-Tarki theorem, it is possible further prove a quantifier elimination procedure for real closed field as Cyril Cohen does in Coq. theory PolyMisc imports Complex-Main ∼∼/src/HOL/Library/Poly-Deriv begin 1 lead coefficient definition lead-coeff :: ′a::zero poly ⇒ ′a where lead-coeff p= coeff p (degree p) 2 Misc lemma smult-cancel : fixes p:: ′a::idom poly assumes c 6=0 and smult : smult c p = smult c q shows p=q 〈proof 〉 lemma dvd-monic: fixes p q :: ′a :: idom poly assumes monic:lead-coeff p=1 and p dvd (smult c q) and c 6=0
منابع مشابه
The Sturm-Tarski Theorem
We have formalised the Sturm-Tarski theorem (also referred as the Tarski theorem): Given polynomials p, q ∈ R[x], the Sturm-Tarski theorem computes the sum of the signs of q over the roots of p by calculating some remainder sequences. Note, the better-known Sturm theorem is an instance of the Sturm-Tarski theorem when q = 1. The proof follows the classic book by Basu et al. [1] and Cyril Cohen’...
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