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arXiv · 2512.07511

Canonical bidirectional typechecking

Zanzi Mihejevs·Jules Hedges
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Abstract

We demonstrate that the checkable/synthesisable split in bidirectional typechecking coincides with existing dualities in polarised System L, also known as polarised $\mu\tilde{\mu}$-calculus. Specifically, positive terms and negative coterms are checkable, and negative terms and positive coterms are synthesisable. This combines a standard formulation of bidirectional typechecking with Zeilberger's `cocontextual' variant. We extend this to ordinary `cartesian' System L using Mc Bride's co-de Bruijn formulation of scopes, and show that both can be combined in a linear-nonlinear style, where linear types are positive and cartesian types are negative. This yields a remarkable 3-way coincidence between the shifts of polarised System L, LNL calculi, and bidirectional calculi.

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Zanzi Mihejevs, Jules Hedges. 2025-12-08. Canonical bidirectional typechecking. https://arxiv.org/abs/2512.07511

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