initial algebra of an endofunctor, terminal coalgebra of an endofunctor

initial algebra of a presentable ∞-monad

natural numbers object, list,

identity type

W-type

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deductive reasoning, deduction

sequent

hypothesis/context/antecedent $\vdash$ conclusion/consequence/succedent

logical framework

deductive system,

natural deduction

sequent calculus

inductive reasoning

induction, recursion

inductive type, higher inductive type

induction over arbitrary well-ordered sets.