Certifying Differential Equation Solutions from Computer Algebra Systems in Isabelle/HOL

The paper states Theorem 1 (Derivative Introduction Theorems) for has-vector-derivative within a set, covering constant, identity, pairing, sum, sine-chain, square-root, and quotient rules with the expected domain provisos (f(t)>0 for sqrt, g(t)≠0 for division) and uses these laws in its certification tactic; it does not derive them from first principles, but treats them as standard library results (HOL-Analysis) . The candidate solution directly proves the same seven items from the limit definition of the within-T derivative, supplying the necessary continuity-from-differentiability lemma and standard first-principles derivatives (sin, sqrt, reciprocal). The results and hypotheses align with the paper’s statements (notably items (e)–(g) and their provisos), and the quotient expression matches the paper’s typed form −f(t)·(1/(g(t))·g′·1/(g(t))) + f′/g(t) . Thus, both are correct and consistent; the paper cites standard laws, while the model gives a constructive limit proof.