As the theoretical foundation of Lagrangian mechanics, Euler–Lagrange equation sets are widely applied in building mathematical models of physical systems, especially in solving dynamics problems. However, their preconditions are often not fully satisfied in practice. Therefore, it is necessary to verify their applications. The purpose of the present work is to conduct such verification by establishing a formal theorem library of Lagrangian mechanics in HOL Light. For this purpose, some basic concepts such as functional variation and the necessary conditions for functional extreme are formalized. Then, the fundamental lemma of variational calculus is formally verified and some new constuctors and destructors are proposed. Finally, the Euler–Lagrange equation set is formalized. To validate the formalization, the formalization results are applied to verify the least resistance problem of gas flow. The present work not only lays a necessary and solid foundation for application involving Lagrangian mechanics but also extends the HOL Light theorem library.

Various types of calculi (Hilbert, Gentzen sequent, resolution calculi, tableaux) for propositional linear temporal logic (PLTL) have been invented. In this paper, a sound and complete loop-type sequent calculus $$\mathbf{G} _\text {L}{} \mathbf{T} $$ for PLTL with the temporal operators “next” and “henceforth always” ( $${\mathbf{PLTL}}^{n,a}$$ ) is considered at first. We prove that all rules of $$\mathbf{G} _\text {L}{} \mathbf{T} $$ are invertible and that the structural rules of weakening and contraction, as well as the rule of cut, are admissible in $$\mathbf{G} _\text {L}{} \mathbf{T} $$ . We describe a decision procedure for $${\mathbf{PLTL}}^{n,a}$$ based on the introduced calculus $$\mathbf{G} _\text {L}{} \mathbf{T} $$ . Afterwards, we introduce a sound and complete sequent calculus $$\mathbf{G} _\text {L}{} \mathbf{T} ^\mathcal {U}$$ for PLTL with the temporal operators “next” and “until”.

This work is a case study of the formal verification and complexity analysis of some famous probabilistic algorithms and data structures in the proof assistant Isabelle/HOL. In particular, we consider the expected number of comparisons in randomised quicksort, the relationship between randomised quicksort and average-case deterministic quicksort, the expected shape of an unbalanced random Binary Search Tree, the randomised binary search trees described by Martínez and Roura, and the expected shape of a randomised treap. The last three have, to our knowledge, not been analysed using a theorem prover before and the last one is of particular interest because it involves continuous distributions.

In complex analysis, the winding number measures the number of times a path (counter-clockwise) winds around a point, while the Cauchy index can approximate how the path winds. We formalise this approximation in the Isabelle theorem prover, and provide a tactic to evaluate winding numbers through Cauchy indices. By further combining this approximation with the argument principle, we are able to make use of remainder sequences to effectively count the number of complex roots of a polynomial within some domains, such as a rectangular box and a half-plane.

A shallow semantical embedding of free logic in classical higher-order logic is presented, which enables the off-the-shelf application of higher-order interactive and automated theorem provers for the formalisation and verification of free logic theories. Subsequently, this approach is applied to a selected domain of mathematics: starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. As a side-effect of this work some (minor) issues in a prominent category theory textbook have been revealed. The purpose of this article is not to claim any novel results in category theory, but to demonstrate an elegant way to “implement” and utilize interactive and automated reasoning in free logic, and to present illustrative experiments.

We describe how we have defined in ACL2 a weak version of the Common Lisp functional apply, which takes a function and list of actuals and applies the function to the actuals. Our version, called apply$, does not operate on functions but on ordinary objects—symbols and lists representing lambda expressions—some of which are interpreted as functions. We define a syntactic notion of “tameness” to identify the interpretable objects. This makes our apply$ weaker than a true second-order functional but we believe apply$ is powerful enough for many uses in ACL2. To maintain soundness and the conservativity of our Definitional Principle we require that certain hypotheses, called “warrants”, be present in any theorem relying on the behavior of apply$ on non-primitives. Within these constraints we can define “functionals” such as sum and foldr which map tame “functions” over lists and accumulate the results. This allows the ACL2 user to avoid defining specialized recursive functions for each such application. We can prove and use general-purpose lemmas about these “functionals.” We describe the formalization, explain how we keep the Definitional Principle conservative, show examples of useful functions using apply$ and theorems about them, sketch the proof that there is a model of any extension of the system using the new primitives, discuss issues arising in making these functions executable, and show some preliminary performance results.

Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF by replacing the linear order of the quantifier prefix with a partial order. We propose to merge the two, by reinterpreting a dependency scheme as a mapping from QBF into DQBF. Our approach offers a fresh insight on the nature of soundness in proof systems for QBF with dependency schemes, in which a natural property called ‘full exhibition’ is central. We apply our approach to QBF proof systems from two distinct paradigms, termed ‘universal reduction’ and ‘universal expansion’. We show that full exhibition is sufficient (but not necessary) for soundness in universal reduction systems for QBF with dependency schemes, whereas for expansion systems the same property characterises soundness exactly. We prove our results by investigating DQBF proof systems, and then employing our reinterpretation of dependency schemes. Finally, we show that the reflexive resolution path dependency scheme is fully exhibited, thereby proving a conjecture of Slivovsky.