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Constructive Logic
Lecture 26: Propositions as Types

The close correspondence between propositions and types allows one to regard types themselves as propositions. Proof terms are promoted to full-fledged entities, and the notion of a constructive proof is then identical with that of a construction of a suitable object. This conception, called ``Propositions as Types,'' is often formulated in the slogan: A proposition is the type of its proofs. We formalize it by strengthening the elimination rule for Sigma types. A noteworthy consequence of this revision is that the type-theoretic version of the Axiom of Choice is constructively provable.

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