# Algebraic composition and refinement of proofs

## Abstract

We present an algebraic calculus for proof composition and refinement. Fundamentally, proofs are expressed at successive levels of abstraction, with the perhaps unconventional principle that a formula is considered to be its own most abstract proof, which may be refined into increasingly concrete proofs. Consequently, we suggest a new paradigm for expressing proofs, which views theorems and proofs as inhabiting the same semantic domain. This algebraic/model-theoretical view of proofs distinguishes our approach from conventional typetheoretical or sequent-based approaches in which theorems and proofs are different entities. All the logical concepts that make up a formal system — formulas, inference rules, and derivations — are expressible in terms of the calculus itself. Proofs are constructed and structured by means of a composition operator and a consequential rule-forming operator. Their interplay and their relation wrt. the refinement order are expressed as algebraic laws.

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