Induction principles for type theories, internally to presheaf categories
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We present new induction principles for the syntax of dependent type theories. These induction principles are expressed in the internal language of presheaf categories. This ensures for free that any construction is stable under substitution. In order to combine the internal languages of multiple presheaf categories, we use Dependent Right Adjoints and Multimodal Type Theory. Categorical gluing is used to prove these induction principles, but it not visible in their statements, which involve a notion of model without context extensions. We illustrate the derivation of these induction principles by the example of type theory with a hierarchy of universes closed under function space and natural numbers, but we expect that our method can be applied to any syntax with bindings. As example applications of these induction principles, we give short and boilerplate-free proofs of canonicity and normalization for our example type theory.
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