Proof-relevance in Bishop-style constructive mathematics
نویسندگان
چکیده
Abstract Bishop’s presentation of his informal system constructive mathematics BISH was on purpose closer to the proof-irrelevance classical mathematics, although a form proof-relevance evident in use several notions moduli (of convergence, uniform continuity, differentiability, etc.). Focusing membership and equality conditions for sets given by appropriate existential formulas, we define certain families proof that provide BHK-interpretation formulas correspond standard atomic first-order theory, within Bishop set theory $(\mathrm{BST})$ , our minimal extension sets. With machinery general sets, this BST is extended complex formulas. Consequently, can associate many $\phi$ ${\texttt{Prf}}(\phi)$ “proofs” or witnesses . Abstracting from examples totalities BISH, notion with proof-relevant equality, Martin-Löf set, special case former, which corresponds identity type intensional $(\mathrm{MLTT})$ Through concepts results facts MLTT its extensions (either axiom function extensionality Vooevodsky’s univalence) be translated into BISH. While standardly understood through translation MLTT, development offers partial converse direction.
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ژورنال
عنوان ژورنال: Mathematical Structures in Computer Science
سال: 2022
ISSN: ['1469-8072', '0960-1295']
DOI: https://doi.org/10.1017/s0960129522000159