3 research outputs found
A First Complete Algorithm for Real Quantifier Elimination in Isabelle/HOL
We formalize a multivariate quantifier elimination (QE) algorithm in the
theorem prover Isabelle/HOL. Our algorithm is complete, in that it is able to
reduce any quantified formula in the first-order logic of real arithmetic to a
logically equivalent quantifier-free formula. The algorithm we formalize is a
hybrid mixture of Tarski's original QE algorithm and the Ben-Or, Kozen, and
Reif algorithm, and it is the first complete multivariate QE algorithm
formalized in Isabelle/HOL
Toward Structured Proofs for Dynamic Logics
We present Kaisar, a structured interactive proof language for differential
dynamic logic (dL), for safety-critical cyber-physical systems (CPS). The
defining feature of Kaisar is *nominal terms*, which simplify CPS proofs by
making the frequently needed historical references to past program states
first-class. To support nominals, we extend the notion of structured proof with
a first-class notion of *structured symbolic execution* of CPS models. We
implement Kaisar in the theorem prover KeYmaera X and reproduce an example on
the safe operation of a parachute and a case study on ground robot control. We
show how nominals simplify common CPS reasoning tasks when combined with other
features of structured proof. We develop an extensive metatheory for Kaisar. In
addition to soundness and completeness, we show a formal specification for
Kaisar's nominals and relate Kaisar to a nominal variant of dL
Structured induction proofs in Isabelle/Isar
Isabelle/Isar is a generic framework for human-readable formal proof documents, based on higher-order natural deduction. The Isar proof language provides general principles that may be instantiated to particular object-logics and applications. We discuss specific Isar language elements that support complex induction patterns of practical importance. Despite the additional bookkeeping required for induction with local facts and parameters, definitions, simultaneous goals and multiple rules, the resulting Isar proof texts turn out well-structured and readable. Our techniques can be applied to non-standard variants of induction as well, such as co-induction and nominal induction. This demonstrates that Isar provides a viable platform for building domain-specific tools that support fully-formal mathematical proof composition