inverse limit construction

This work presents the first complete formalization in Lean 4, based on mathlib, of Dana Scott’s 1972 theory of continuous lattices and its application to modeling the untyped λ-calculus. The project rigorously reproduces the 43 core results from the first four sections of Scott’s original paper, covering essential constructions such as T₀-space embeddings, the Scott topology, the way-below relation, function spaces, and inverse limits, while also incorporating Milner’s corrections to the original proofs. By introducing foundational infrastructure—including bases of Scott-open sets, step functions, towers of function spaces, and the i_∞/j_∞ mapping pair—the formalization establishes the self-embedding theorem D_∞ ≅ [D_∞ → D_∞] under classical logic (with the axiom of choice), propositional extensionality, and quotient type soundness assumptions. All results are verified machine-checked without any use of “sorry”.

Constructive mathematics faces a fundamental limitation: even when aiming to characterize uncountable objects—such as the continuum—all syntactically enumerable constructive methods, whether employing diagonalization or not, can only generate countable fragments within any closed formal system. Method: The authors introduce the notion of a “constructive fractal boundary” to formalize the inherent countability ceiling imposed on constructive processes at the metatheoretic level; they further define “fractal countability,” establishing a novel framework for fine-grained analysis of definability beyond classical recursion—without presupposing an uncountable totality. Contribution/Results: The central result establishes that the continuum is not a constructively realizable object, but rather an asymptotic limit of expressive capacity for formal systems. Integrating metatheory of formal systems, recursion theory, and constructive semantics, the paper rigorously proves that no syntactically enumerable constructive system can fully capture the continuum.

This paper investigates the computational nature of result sequences generated by fixed-subroutine iteration in infinite-loop algorithms, focusing on their cyclicity, reverse-limit behavior, and non-determinism within the Weihrauch complexity framework. Methodologically, it integrates Weihrauch reducibility, the injective recursion theorem, and the infinite independent choice theorem, while systematically comparing parallelization with the diamond operator composition. The key contribution is the formal introduction and analysis of the *inverse limit operator*, which is proven monotone but not closed under composition; it is shown to be closed under the weak König lemma and a broad class of non-deterministically computable problems. Crucially, the inverse limit operator is strictly stronger than the combination of parallelization and the diamond operator, and exhibits well-behaved properties on important problem classes—including single-valued and Turing-degree-related problems—thereby establishing a novel paradigm for characterizing the structural complexity of iterative computation.

This paper addresses the lack of consensus on the definition of “computable topological bases” in computable analysis, systematically clarifying the logical relationships among various notions of computable second-countability and their connections to the Sierpiński representation and computable metrizability. By integrating tools from represented spaces theory, computably enumerable bases, strong computable regularity, and open choice problems, the authors prove that several ostensibly distinct base concepts become equivalent under the assumption of computable enumerability—thereby establishing a robust definition of computable second-countable spaces. Key contributions are: (1) a unifying compatibility framework and hierarchy relating diverse approaches to computable topology; (2) an effective metrization theorem precisely characterizing represented spaces embeddable into computable metric spaces; and (3) a deep connection between non-surjective open choice and effective second-countability.

This work investigates the creative space of mathematical proofs under constraints, with a particular focus on the impact of non-constructive reasoning. We introduce a strategy ablation methodology that integrates our custom-built Meno automated formalization tool with Goedel Prover embeddings to systematically explore both formal and informal proof spaces for foundational theorems from *Analysis I* within the Lean theorem prover. Our experiments successfully generate a novel class of machine-produced proofs, revealing that these proofs cluster along low-dimensional submanifolds in a high-dimensional representation space and significantly diverge from human-constructed proof trajectories. This study provides the first quantitative characterization of the structural differences between machine-generated and human proofs.

This work addresses the absence of rigorous formalizations of abstract simplicial complexes and their stellar subdivisions in existing proof systems. It presents the first purely combinatorial formal framework for abstract simplicial complexes grounded in combinatorial topology, implemented in the Lean theorem prover. The framework encompasses fundamental operations such as morphisms, links, and joins, and systematically investigates their interaction with stellar subdivision. Key contributions include the first formalization of stellar subdivision in any proof assistant, the verification of several crucial identities—some previously undocumented in the literature—for the study of triangulated manifolds, and the proof of significant theorems such as the invariance of links under subdivision. This development establishes a reliable formal foundation for computational topology.

This work addresses the limitation of traditional introductory computer science curricula, which often emphasize isolated knowledge points while neglecting the underlying proof techniques and abstract structures essential for cultivating computational thinking in beginners. To remedy this, the paper proposes a novel pedagogical paradigm centered on universal proof strategies and abstract frameworks, using the transitive closure of relations as a representative case study. By integrating tools such as the Kleene star, quantale theory, and closure operators over complete lattices, the approach constructs a cohesive bridge linking logic, algebra, and computational reasoning. This method yields a generalizable instructional framework that significantly enhances students’ structural understanding and analytical capacity regarding foundational concepts.

This work presents the first assumption-free formalization of Cauchy real numbers in Cubical Agda, circumventing reliance on the axiom of choice, setoid bookkeeping, or explicit universe-level management—issues that commonly hinder constructive real number constructions in intuitionistic mathematics. Building upon the higher inductive-inductive types introduced in Homotopy Type Theory, the construction leverages Cubical Agda’s native support for higher inductive types to yield a fully type-checked, non-vacuous, and postulate-free development of the reals. This approach not only resolves longstanding challenges related to redundancy and universe complexity but also establishes a robust foundation for machine-verified constructive analysis.

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 work proposes a systematic formalization of all published mathematical knowledge into a machine-verifiable, continuously evolving structured knowledge base, addressing the challenges of scalability and organization. Taking dilatations of categories in categorical algebra as the first case study, the project integrates interactive theorem proving, dependent type theory, and category theory to frame the complete formalization of mathematics as a universal reasoning benchmark. By constructing a formal prototype of category dilatations, the study demonstrates the feasibility of this approach in expressing complex algebraic structures, thereby establishing both an architectural foundation and a practical pathway toward a large-scale, interconnected, and extensible database of formalized mathematics.