construct mathematical proofs

Defining mathematical concepts formally remains a critical bottleneck in interactive theorem proving: steep learning curves hinder newcomers, and undergraduate-level formalization progresses slowly. This paper investigates the generality, readability, and type-system compatibility of definitions, using Lean’s mathlib as an empirical foundation. We systematically analyze hundreds of equivalent definitions across diverse mathematical domains, evaluating them via usability metrics—theorem verification success rate, proof conciseness, and interface orthogonality. We identify three key determinants of definition quality: abstraction level, constructive strength, and interface granularity; from these, we distill reusable design principles. Furthermore, we contrast definition strategies in computer algebra systems (CAS) and, for the first time, establish a cross-system formal definition design guide. Our framework significantly improves the efficiency of standardized knowledge construction and long-term collaborative sustainability in libraries such as mathlib.

This paper addresses two key challenges in foundational programming theory: the mathematical complexity of basic programming concepts and the reliance of formal verification on redundant axioms. To resolve these, we propose PRISM—a minimalist programming theory grounded solely in naive set theory. PRISM introduces only one primitive relation, one initial set, and three fundamental operations (selection, composition, and restriction), without assuming any axioms. It unifies program semantics and specifications within a single set-theoretic framework, defining correctness, specialization, and refinement via standard subset inclusion. All core results—including over thirty program properties and classical “programming laws”—are fully mechanized and formally verified in Isabelle/HOL. Our contributions are threefold: (1) the first axiom-free programming theory framework; (2) a unified semantic–specification representation for programs; and (3) an open-source, reproducible, and extensible library of machine-checked proofs.

This study addresses the problem of systematically constructing logical systems corresponding to program abstractions to support formal reasoning. By associating logical systems with finite abstract domains, the work proposes a general method: for a given abstract lattice, it constructs a logic whose Lindenbaum–Tarski algebra is isomorphic to the abstraction, and derives corresponding axioms and inference rules. This approach establishes, for the first time, a systematic connection between abstract interpretation and proof theory as well as algebraic logic, enabling logical modeling of non-Cartesian abstractions such as octagons. The resulting logical connectives and inference systems preserve the concretization map and, under suitable conditions, satisfy soundness and completeness. The framework naturally extends to Cartesian products, multi-variable settings, and non-Cartesian abstract domains.

This work investigates the goal-directed generation of mathematically meaningful theorems—or lemmas suitable for automated proof—from a given set of axioms. To this end, it introduces a novel approach grounded in the propositions-as-types paradigm, which systematically partitions the space of proof terms according to inductive levels and integrates proof-term enumeration with compression techniques, including separation-based reduction, DAG compression, and combinatory logic. This framework enables the efficient construction and compact representation of proof structures. Experimental evaluation on a fragment of Metamath’s set.mm library demonstrates that the method successfully produces nontrivial and semantically relevant theorems, thereby confirming its feasibility and advantages in the context of automated theorem discovery.

This paper addresses the conceptual fragmentation in sequent-style proof systems across modal, temporal, intuitionistic, conditional, and substructural logics, as well as the ill-defined distinction between “internal” and “external” calculi. We propose a unified classification framework grounded in the fundamental data structure of sequents. By systematically analyzing extant sequent calculi, we develop a hierarchical taxonomy and conduct meta-theoretic analysis of upward and downward logical translations. Crucially, we formally define—and thereby eliminate—the spurious internal/external distinction for the first time. We prove that upward translation preserves structural integrity, whereas downward translation is generally infeasible. Our results establish a principled comparability benchmark and formal assessment toolkit for cross-logical proof systems, advancing the unified modeling of structural proof theory and enabling robust automation support.

This work addresses the challenge of automatically synthesizing provably correct, recursion-free programs from specifications containing non-computable symbols while ensuring the completeness of the synthesis method. The authors extend the superposition calculus to the domain of program synthesis and refine its inference rules to overcome incompleteness in certain cases. Their proposed system achieves, for the first time within a superposition-based framework, synthesis completeness under a realizability assumption: whenever a computable program satisfying the specification exists, it is guaranteed to be synthesized. By integrating automated theorem proving, formal specification reasoning, and realizability analysis, the approach offers both soundness and theoretical completeness.

Traditional formal mathematics emphasizes rigorous proof and mechanical verification, yet its complex tools and high entry barriers hinder accessibility and effective idea exchange for most mathematical practitioners. This work proposes the “free approach”—a novel paradigm centered on communicability and usability that departs from certification-centric frameworks by eliminating the mandatory requirement to formally verify every detail. Grounded in Alonzo logic—a practical variant of simple type theory—the approach establishes both theoretical foundations and implementation mechanisms. The study demonstrates that the free approach aligns more closely with actual mathematical practice, effectively serving a broad community of mathematicians, and calls for collective development of supporting logical systems, toolchains, knowledge repositories, and educational infrastructure.

This work proposes a parameterized, unified realizability framework that overcomes the limitations of traditional realizability interpretations, which require explicit witnesses for existential quantifiers and struggle to uniformly handle atomic formulas alongside quantified statements. By abstracting and formally characterizing the semantic treatment of atomic formulas, the framework encompasses a wide spectrum of classical and modern realizability interpretations. It is shown to be compatible with various logical systems, including Heyting arithmetic, where its expressive power and consistency are rigorously verified. This approach enables a systematic integration and comparative analysis of diverse realizability methods within a single coherent setting.

This work presents a systematic investigation of the interpolation property across classical, intuitionistic, modal, and substructural logics, proposing a unified, constructive, modular, and syntax-driven methodology that integrates Maehara’s and Pitts’ classical techniques into a general proof-theoretic framework. By uncovering a structural correspondence between interpolation and well-behaved proof systems, the study not only establishes the existence of interpolation theorems for a wide range of logics but also delivers a reusable blueprint for constructing interpolation proofs within modern formal proof systems. This approach emphasizes the role of syntactic structure in enabling modular and scalable interpolation results, thereby advancing the theoretical understanding and practical applicability of interpolation in diverse logical settings.