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José A. Alonso

Verified program extraction in number theory: The fundamental theorem of arithmetic and relatives. ~ Franziskus Wiesnet. arxiv.org/abs/2504.03460 #ITP #Minlog #Haskell #FunctionalProgramming #Math

arXiv.orgVerified Program Extraction in Number Theory: The Fundamental Theorem of Arithmetic and RelativesThis article revisits standard theorems from elementary number theory through a constructive, algorithmic, and proof-theoretic lens, within the theory of computable functionals. Key examples include Bèzout's identity, the fundamental theorem of arithmetic, and Fermat's factorization method. All definitions and theorems are fully formalized in the proof assistant Minlog, thereby laying the foundation for a comprehensive formal framework for number theory within Minlog. While formalization ensures correctness, our primary emphasis is on the computational content of proofs. Leveraging Minlog's built-in program extraction, we obtain executable terms that are exported as Haskell code. Efficiency of the extracted programs plays a central role. We show how performance considerations influence even the initial formulation of theorems and proofs. In particular, we compare formalizations based on binary encodings of natural numbers with those using the traditional unary (successor-based) representation. We present several core proofs in detail and reflect on the challenges that arise from formalization in contrast to informal reasoning. The complete formalisation is available online and linked for reference. Minlog's tactic scripts are designed to follow the structure of natural-language proofs, allowing each derivation step to be traced precisely and helping to bridge the gap between formal and classical mathematical reasoning.
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