A Zoo of Continuity Properties in Constructive Type Theory

Continuity principles stating that all functions are continuous play a central role in some schools of constructive mathematics. However, there are different ways to formalise the property of being continuous in constructive foundations. We analyse these continuity properties from the perspective of constructive reverse mathematics. We work in constructive type theory, which can be seen as a minimal foundation for constructive reverse mathematics. We treat continuity of functions F : (Q → A) → R, i.e. with question type Q, answer type A, and result type R. Concretely, we discuss continuity defined via moduli, making the relevant list L : LQ of questions explicit, dialogue trees, making the question-answer process explicit as inductive tree, and tree functions, making the question-answer process explicit as function. We prove equivalences where possible and isolate necessary and sufficient axioms for equivalence proofs. Many of the results we discuss are already present in the works of Hancock, Pattinson, Ghani, Kawai, Fujiwara, Brede, Herbelin, Escardó, and others. Our main contribution is their formulation over a uniform foundation, the observation that no choice axioms are necessary, the generalisation to arbitrary types from natural numbers where possible, and a mechanisation in the Coq/Rocq proof assistant.