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The principle of propositions-as-types (or formulas-as-types), also known as the Curry-Howard correspondence, is the key idea for viewing (intuitionistic) type theories as logical systems and to apply type theories (and constructive mathematics in general) to computer science. Suppose U is the type of all propositions, then U is also the type of all types by ...


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$\text{absurd} : (A : \text{U}) \to 0 \to A$ and $\text{elim} : (A : 0 \to \text{U}) \to (x : 0) \to A\,x$ are equivalent. To go right, use $\text{absurd}\,(A\,x)\,x$. To go left, use $\text{elim}\,(\lambda\,x.\,A)\,x$. Also, both types are propositions because of the $0$-s in domains. There's not much reason to assume or use $\text{elim}$ instead of $\text{...


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