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The homotopy interpretation of constructive dependent type theory, the univalence axiom, higher inductive types, internal languages of higher toposes, univalent foundations for mathematics, and implementations of such theories in proof assistants.
10
votes
Appearance of proof relevance in "ordinary mathematics?"
Proof mining (which even has a short Wikipedia article!), and area in large part developed by Kohlenbach, is mentioned briefly in the comments, and I thought it deserves a bigger mention. Roughly spea …