The algebraic $\lambda$-calculus is a conservative extension of the ordinary $\lambda$-calculus

Axel Kerinec; Lionel Vaux Auclair

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arXiv · 2305.01067

The algebraic $\lambda$-calculus is a conservative extension of the ordinary $\lambda$-calculus

Axel Kerinec·Lionel Vaux Auclair

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Abstract

The algebraic $\lambda$-calculus is an extension of the ordinary $\lambda$-calculus with linear combinations of terms. We establish that two ordinary $\lambda$-terms are equivalent in the algebraic $\lambda$-calculus iff they are $\beta$-equal. Although this result was originally stated in the early 2000's (in the setting of Ehrhard and Regnier's differential $\lambda$-calculus), the previously proposed proofs were wrong: we explain why previous approaches failed and develop a new proof technique to establish conservativity.

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Axel Kerinec, Lionel Vaux Auclair. 2023-05-01. The algebraic $\lambda$-calculus is a conservative extension of the ordinary $\lambda$-calculus. https://arxiv.org/abs/2305.01067

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