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’s work in Coq [2]. With the Sturm-Tarski theorem proved, it is possible to further build a quantifier elimination procedure for real numbers as Cohen did in Coq. Another application of the Sturm-Tarski theorem is to build sign determination procedures for polynomials at real algebraic points, as described in our formalisation of real algebraic numbers [3]. theory PolyMisc imports HOL−Computational-Algebra.Polynomial-Factorial begin lemma coprime-poly-0 : assumes coprime p q shows poly p x 6=0 ∨ poly q x 6=0 by (metis assms poly-1 poly-eq-0-iff-dvd semiring-gcd-class.gcd-greatest-iff zero-neq-one) lemma smult-cancel : fixes p:: ′a::idom poly assumes c 6=0 and smult : smult c p = smult c q shows p=q proof − have smult c (p−q)=0 using smult by (metis diff-self smult-diff-right) thus ?thesis using 〈c 6=0 〉 by auto qed 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 shows p dvd q using assms