相关论文: Conversion of HOL Light proofs into Metamath
An Isabelle/HOL formalisation of G\"odel's two incompleteness theorems is presented. The work follows \'Swierczkowski's detailed proof of the theorems using hereditarily finite (HF) set theory. Avoiding the usual arithmetical encodings of…
As the development of formal proofs is a time-consuming task, it is important to devise ways of sharing the already written proofs to prevent wasting time redoing them. One of the challenges in this domain is to translate proofs written in…
Church's Higher Order Logic is a basis for influential proof assistants -- HOL and PVS. Church's logic has a simple set-theoretic semantics, making it trustworthy and extensible. We factor HOL into a constructive core plus axioms of…
This paper describes a formal theory of smooth vector fields, Lie groups and the Lie algebra of a Lie group in the theorem prover Isabelle. Lie groups are abstract structures that are composable, invertible and differentiable. They are…
When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theorems from Maskin's 2004 survey article on Auction Theory using the Isabelle/HOL…
$\alpha$Check is a light-weight property-based testing tool built on top of $\alpha$Prolog, a logic programming language based on nominal logic. $\alpha$Prolog is particularly suited to the validation of the meta-theory of formal systems,…
We present Proof-of-Perception (PoP), a tool-using framework that casts multimodal reasoning as an executable graph with explicit reliability guarantees. Each perception or logic node outputs a conformal set, yielding calibrated, stepwise…
The need for formal definition of the very basis of mathematics arose in the last century. The scale and complexity of mathematics, along with discovered paradoxes, revealed the danger of accumulating errors across theories. Although,…
Building on our previous work on hybrid polyadic modal logic we identify modal logic equivalents for Matching Logic, a logic for program specification and verification. This provides a rigorous way to transfer results between the two…
The ever-growing complexity of mathematical proofs makes their manual verification by mathematicians very cognitively demanding. Autoformalization seeks to address this by translating proofs written in natural language into a formal…
Structural proof theory is praised for being a symbolic approach to reasoning and proofs, in which one can define schemas for reasoning steps and manipulate proofs as a mathematical structure. For this to be possible, proof systems must be…
When prompted with a few examples and intermediate steps, large language models (LLMs) have demonstrated impressive performance in various reasoning tasks. However, prompting methods that rely on implicit knowledge in an LLM often generate…
The formalisation of mathematics is continuing rapidly, however combinatorics continues to present challenges to formalisation efforts, such as its reliance on techniques from a wide range of other fields in mathematics. This paper presents…
Isabelle is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a meta-logic (or `logical framework') in…
We present a formalization of higher-order logic in the Isabelle proof assistant, building directly on the foundational framework Isabelle/Pure and developed to be as small and readable as possible. It should therefore serve as a good…
Principia Logico-Metaphysica contains a foundational logical theory for metaphysics, mathematics, and the sciences. It includes a canonical development of Abstract Object Theory [AOT], a metaphysical theory (inspired by ideas of Ernst…
Dependently typed lambda calculi such as the Edinburgh Logical Framework (LF) are a popular means for encoding rule-based specifications concerning formal syntactic objects. In these frameworks, relations over terms representing formal…
Previous results on proving confluence for Constraint Handling Rules are extended in two ways in order to allow a larger and more realistic class of CHR programs to be considered confluent. Firstly, we introduce the relaxed notion of…
We implement a user-extensible ad hoc connection between the Lean proof assistant and the computer algebra system Mathematica. By reflecting the syntax of each system in the other and providing a flexible interface for extending…
Lighthouse projects such as CompCert, seL4, IronFleet, and DeepSpec have demonstrated that full verification of entire systems is feasible by establishing a refinement relation between an abstract system specification and an executable…