inverse-limit formalization

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”.

This work addresses the longstanding gap in formal mathematics concerning the continuous functional calculus for C*-algebras. Using Lean 4 and the Mathlib library, we present the first fully verified formalization of this theory in any proof assistant: we rigorously define the continuous functional calculus on arbitrary C*-algebras, construct a framework for continuous maps from compact subsets of ℂ to bounded operators, and formally verify its fundamental properties—including the spectral mapping theorem, algebra homomorphism property, and continuity. Our design balances mathematical naturalness with seamless integration into Mathlib’s existing infrastructure; all results have been merged into the main Mathlib repository. This formalization establishes a foundational cornerstone for the mechanized development of C*-algebra theory and spectral theory, while providing highly reusable interfaces that significantly facilitate subsequent formalizations—such as spectral decomposition and the classification of normal operators.

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 conceptual distinction—and need for unified modeling—between verifiability (as an intrinsic property) and verification procedures (as methods) in topology. Method: We introduce the novel notion of a “machine space” and construct a weak exponential space Σ^{Σ^G} as a topological realization of the Sierpiński exponential Σ^X, establishing evaluation maps via frame semantics and generator composition. Contribution/Results: We characterize, for the first time, obstructions to the existence of exponential objects using weak exponential structure. Moreover, we provide a purely topological realization of universal quantification over compact spaces—proving its equivalence to finite-step decidability—and thereby expose a fundamental connection between compactness and computational feasibility. Our framework unifies point-set topology, domain theory, and Escardó’s algorithmic topology, confirming that machine spaces always exist and that their non-contractibility characterizes the obstruction to exponentiation.

In dependent type theory, controlling the granularity of definition unfolding has long posed a dilemma: global unfolding renders proofs brittle, while manual annotation sacrifices automation and robustness. This paper introduces a novel localized unfolding mechanism grounded semantically in extension types, whereby definitions remain inert by default and admit on-demand, local, context-sensitive selective unfolding. Departing from traditional global toggle paradigms, we design a core calculus within homotopy type theory and implement a formally verified prototype in the cooltt proof assistant. Theoretically, we establish a normalization theorem ensuring computational soundness. Empirically, our approach significantly improves proof stability, effectively curbs target-term size explosion, and reconciles expressive reasoning power with maintainability—demonstrating both foundational advancement and practical viability.

This work addresses the challenge of proving definitional inversion properties—such as injectivity and non-confusion—for type constructors in dependent type theories without relying on normalization. To this end, it introduces a novel metatheoretic framework grounded in domain theory, which for the first time establishes the injectivity of type constructors rigorously within a non-normalizing system featuring both η-laws and the type-in-type rule. The approach entirely dispenses with normalization assumptions, has been validated in a minimal type theory, and shows promise for extension to practical non-normalizing systems such as Idris and Lean, thereby opening a new avenue for metatheoretic investigations of dependent type theories.

This work addresses the systematic construction, within category theory, of a new class of categories in which every sieve element belonging to a given “center”—comprising morphisms and sieves—admits a unique and functorial factorization through its associated morphism. To achieve this, the authors introduce the notion of a *dilatation* category and present the first complete formalization of this construction together with its core theorems in Lean 4. Built upon the Mathlib library, the project establishes a precise correspondence between rigorous mathematical definitions and their formal proofs, accompanied by a detailed dictionary mapping mathematical concepts to their code implementations. This effort not only guarantees the logical correctness of the underlying theory but also lays a reusable foundation for machine-checked research in category theory and related fields.

This work addresses the limitations of existing Taylor expansion theory, which struggles to apply to classical web-based models of linear logic such as Köthe spaces and finiteness spaces, particularly when dealing with non-positive coefficients and partial summation structures. The paper introduces a general web-based semantic framework that accommodates partial summation and, for the first time, extends Taylor expansion theory to settings involving non-positive coefficients. This unified approach encompasses coherence spaces, probabilistic coherence spaces, finiteness spaces, and Köthe spaces. By integrating semantic tools from linear logic, differential λ-calculus, sequence space theory, and absolute convergence analysis of formal power series, the authors demonstrate that all major web-based models satisfy a generalized form of Taylor expansion, thereby broadening the mathematical foundations and applicability of differential program semantics.

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 presents the first complete formalization of Wolstenholme’s theorem in Lean 4, establishing that for any prime \( p \geq 5 \), the congruence \( \binom{2p}{p} \equiv 2 \pmod{p^3} \) holds. The proof proceeds by expanding the shifted factorial product up to terms of order \( p^2 \), identifying its quadratic coefficient as the second elementary symmetric polynomial, and demonstrating that this coefficient is divisible by \( p \) using the fact that power sums vanish modulo \( p \). Built entirely on the Mathlib library without any unproven assumptions (i.e., no `sorry`), the formalization comprises nine lemmas and approximately 800 lines of code. It leverages a combination of relational analogy reasoning and human-guided exploration to uncover the critical proof pathway, marking the first fully verified formalization of this classical number-theoretic result in an interactive theorem prover.