HOL Light Theorem Prover

• HOL Light is an interactive proof assistant based on classical higher-order logic with a minimal, trusted kernel for reliable formal verification.
• It supports diverse formalization paradigms including deep embeddings, object-level logic encodings, and AI/ATP-driven automation.
• Its architecture leverages OCaml and a hierarchical tactic system to enable scalable and precise formal proofs in advanced mathematical analysis.

The HOL Light theorem prover is an interactive proof assistant implementing classical higher-order logic (Church’s simple type theory) in the OCaml language. It is distinguished by an extremely small, trusted kernel—comprising only three axioms and ten primitive inference rules—upon which all derived mathematics and automation are constructed. This architecture has enabled the formal verification of substantial mathematical theorems while maintaining maximal reliability and extensibility. HOL Light supports a wide variety of formalization paradigms, including deep embeddings of mathematics, direct object-level logic encodings, mechanized analysis, automation via AI/ATP integration, and recent advances in machine learning–driven theorem proving.

1. Logical Foundations and System Architecture

HOL Light implements Church’s higher-order logic, with terms and types corresponding directly to simply typed λ-calculus. The trusted kernel, about 400 lines of OCaml, allows theorem construction only via a fixed set of inference rules on sequents, terms, and types. The native logic supports classical reasoning, polymorphism, and functional abstraction.

Key features of this architecture include:

• Reliability: Proofs are machine-checked against a minimal set of core rules; only the kernel need be trusted.
• Extensibility: New concepts (types, constants, functions, inductive predicates) are conservatively added at the ML level above the kernel; the kernel itself remains unchanged (Hales, 2014).
• Automation: Above the kernel, a hierarchical tactic and decision-procedure system supports interactive and automated proof development (Kaliszyk et al., 2013).

The data structures and proof objects are OCaml terms, where each theorem is a value of type thm. No external procedure, tactic, or code outside the trusted kernel can declare a theorem as such. This minimalist approach to trusted computing base has been verified in multiple ways, including self-verification of the kernel and compilation to machine code via the verified CakeML compiler (Hales, 2014).

2. Logical Frameworks and Mathematical Embeddings

HOL Light’s logic is fundamentally classical simple type theory with ML-style polymorphism. The core datatypes include:

• Base types: bool, num, int, real
• Type constructors: product, sum, function, sets, lists
• Terms: λ-terms, constants, applications, abstractions

Derived infrastructure allows the definition and use of recursive and inductive data types (e.g., lists, trees), Boolean-valued set-forming operations, and abstract algebraic and topological constructs (Paulson, 2018). For example, Euclidean spaces ℝⁿ are encoded as function types A → real with a finite indexing type A, enabling the formalization of high-dimensional analysis and geometry.

The proof framework is maximally modular, supporting direct deep embeddings (defining formulas as data and specifying semantics), and object-level reasoning inside encoded logics. Inductive definitions and recursive rules underpin the construction of complex mathematical objects and formalization of entire branches of mathematics, such as real analysis, topology, measure theory, and combinatorics (Hales, 2014, Paulson, 2018).

3. Methodologies for Formal Proof and Automation

Proof development in HOL Light is conducted interactively, using a sophisticated tactic language in OCaml. Users break goals into subgoals via tactics, with the justification of each step back-propagated to the original theorem statement. Decision procedures (for e.g., linear arithmetic, ring equalities) and rewriting engines augment the user-driven tactic approach (Hales, 2014). The MESON tactic provides first-order automation via model elimination, and tactics like REWRITE_TAC and SIMP_TAC facilitate equational reasoning and basic proof search.

For more advanced automation, HOL Light integrates with external ATPs (Vampire, E, Z3), leveraging sophisticated premise selection powered by machine learning. The HOL(y)Hammer system extends this by learning which theorems are most useful for premise selection in automated proving, running many ATP and decision-procedure backends in parallel (Kaliszyk et al., 2013). Efficient proof reconstruction is performed when ATPs find proofs, so that the resulting HOL theorems are still kernel-checked.

Recent machine learning–driven systems, such as HOList and DeepHOL, frame the proof process as a reinforcement learning or imitation learning problem, learning policies over tactics, arguments, and proof-state representations to guide automated proof search in the native HOL Light environment (Bansal et al., 2019). These methods have achieved automated proof success rates on significant benchmarks, demonstrating that a combination of data-driven and symbolic techniques is effective in higher-order theorem proving.

4. Advanced Embedding and Object-Level Reasoning

HOL Light enables systematic "object-level" reasoning by internalizing entire logical frameworks within its higher-order logic. This is realized via layered infrastructure for inductive predicates (e.g., turnstile relations for sequent calculi), metatheoretic rule specification, and associated tactics for forward and backward chaining (Papapanagiotou et al., 2021).

For example, the encoding of intuitionistic, linear, or modal logics is accomplished by:

• Defining formula data types (e.g., Prop, form)
• Inductive predicates for provability (e.g., |- : Prop multiset -> Prop -> bool)
• Automated matching and application of rules via multiset AC-matching and unification
• Extraction facilities for computational content (Curry–Howard, λ-terms, concurrent processes)

The system supports generic tactics for sequent calculus and related proof systems, allowing for flexible and efficient interactive reasoning across propositional, modal, and computational logics.

5. Case Studies: Formalized Mathematical Analysis

HOL Light has served as the basis for formalizing complex mathematical and engineering analysis, notably including:

• Laplace and Fourier transforms: Comprehensive formalization of definitions, properties (existence, linearity, frequency shifting, differentiation), and case studies of engineering systems. Definitions follow mathematical notation, and proofs rerun the full analytic machinery in higher-order logic, necessitating the development of auxiliary lemmas for improper integrals, vector-valued functions, and handling of side-conditions (Rashid et al., 2017).
• Partial differential equations: Example: formalization of the one-dimensional heat equation using higher-real-derivative operators, proof of linearity and scaling of the heat operator, solution by separation of variables (Fourier sine expansion), and formal proof of infinite-series convergence. Side-conditions (differentiability, integrability) are tracked explicitly, and boundary/initial conditions are handled with the same full formality (Deniz et al., 2022).

These formalizations demonstrate both the expressive power of HOL Light's multivariate analysis libraries and the necessity of augmenting the standard library for advanced applications.

6. Modal and Non-Classical Logics Mechanized

The modal logic ecosystem in HOL Light includes both deep embeddings and “shallow” sequent calculus tactics, enabling mechanized proofs of adequacy (soundness and completeness) for modal systems K, T, K4, and GL. The HOLMS library builds a parametric meta-framework, jointly modeling syntax, semantics (Kripke frames), provability predicates, and constructive countermodel search (Bilotta, 11 Jun 2025).

For Gödel–Löb’s provability logic (GL), HOL Light supports a full formalization of the modal metatheory, including the proof of the canonical completeness theorem over irreflexive, transitive finite models, development of labelled sequent calculus tactics (G3KGL), and extraction of counterexamples for non-theorems (Maggesi et al., 2021, Maggesi et al., 2022, Bilotta, 11 Jun 2025). The strategy adopted explicitly mechanizes maximal-consistent-list argumentation over the subformula universe for each given formula, avoiding the non-compactness pitfall by restricting to finite subformulas, and generalizes completeness by bisimulation invariance to arbitrary infinite carriers.

7. Evaluation, Automation, and Library Organization

Scaling to large mathematical libraries necessitates both intelligent automation and infrastructure-level analyses. The theorem-mining methodology constructs and analyzes the HOL inference graph, ranking implicit lemmas by utility metrics (uses, depth, size, centrality) in order to surface new, effective auxiliary facts for automated proving (Kaliszyk et al., 2013). This improves proof search both in premise selection and direct ATP-driven re-proving, advancing automation rates in large libraries by up to 18% in optimistic scenarios.

In practice, built-in tactics close 40–50% of core analysis goals, with full formalizations (e.g., the Flyspeck Kepler proof) checking in under three hours on modern hardware (Hales, 2014). Machine learning advice and ATP integration further extends automation reach. Local and online services such as HOL(y)Hammer provide turn-key, project-level automated proving, with web, Emacs, and command-line interfaces (Kaliszyk et al., 2013).

• “Developments in Formal Proofs” (Hales, 2014)
• “Formalizing Mathematics In Simple Type Theory” (Paulson, 2018)
• “Formalization of Transform Methods using HOL Light” (Rashid et al., 2017)
• “On the Formalization of the Heat Conduction Problem in HOL” (Deniz et al., 2022)
• “HOList: An Environment for Machine Learning of Higher-Order Theorem Proving” (Bansal et al., 2019)
• “Object-Level Reasoning with Logics Encoded in HOL Light” (Papapanagiotou et al., 2021)
• “A formal proof of modal completeness for provability logic” (Maggesi et al., 2021)
• “Mechanising Gödel-Löb provability logic in HOL Light” (Maggesi et al., 2022)
• “Growing a Modular Framework for Modal Systems- HOLMS: a HOL Light Library” (Bilotta, 11 Jun 2025)
• “HOL(y)Hammer: Online ATP Service for HOL Light” (Kaliszyk et al., 2013)
• “Lemma Mining over HOL Light” (Kaliszyk et al., 2013)