The simple semantics for Coppo-Dezani-Sallé types
The Coppo-Dezani-Sallé type-language has "∩" (intersection) and "∩" (universal type), besides the usual "→" (exponentiation). Coppo, Dezani and Sallé have presented formal rules for assigning types to type-free λ-terms, and have proved metatheorems which give their system significance and interest. (Sallé 1978, Coppo et al. 1981.)
But no precise semantics has been given for their system yet, though the authors have made it fairly obvious informally what semantics they intended.
The present paper defines a precise semantics in an arbitrary model of type-free λ-calculus. The rules of Coppo, Dezani and Sallé turn out to be incomplete with respect to this semantics, but they become complete when three extra rules (two trivial, one not) are added. The completeness proof uses the term-model only.
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