This correspondence generalises to encompass intuitionist arithmetic (Heyting arithmetic), which requires an extension from propositional to predicate logic

Curry and Feys (1958) extended the correspondence idea to one between type theory and Gentzen’s sequent calculus. In the paper already cited, circulated in 1969, but only published in a volume in a Festschrift for Curry in 1980, W.A. Howard (1969) deepened the CH correspondence by demonstrating a correspondence between intuitionistic sequent form natural deduction and type theory in \(\lambda\)-calculus format, generalising to encompass intuitionist arithmetic- ‘Heyting arithmetic’ (HA)- (thus r