The logic of first order intuitionistic type theory with weak sigma- elimination

Journal of Symbolic Logic 56 (2):467-483 (1991)
Abstract
Via the formulas-as-types embedding certain extensions of Heyting Arithmetic can be represented in intuitionistic type theories. In this paper we discuss the embedding of ω-sorted Heyting Arithmetic HA ω into a type theory WL, that can be described as Troelstra's system ML 1 0 with so-called weak Σ-elimination rules. By syntactical means it is proved that a formula is derivable in HA ω if and only if its corresponding type in WL is inhabited. Analogous results are proved for Diller's so-called restricted system and for a type theory based on predicate logic instead of arithmetic
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