### One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations [chapter]

Murdoch J. Gabbay, Dominic P. Mulligan
2008 Lecture Notes in Computer Science
The Curry-Howard correspondence connects Natural Deduction derivation with the lambda-calculus. Predicates are types, derivations are terms. This supports reasoning from assumptions to conclusions, but we may want to reason 'backwards' from the desired conclusion towards the assumptions. At intermediate stages we may have an 'incomplete derivation', with 'holes'. This is natural in informal practice; the challenge is to formalise it. To this end we use a one-and-a-halfth order technique based
more » ... nominal terms, with two levels of variable. Predicates are types, derivations are terms -and the two levels of variable are respectively the assumptions and the 'holes' of an incomplete derivation. A Technical appendix We will use the following fact without comment: Lemma 22 If Γ r is closed (so Γ mentions no unknowns and the freshness context is empty) and Γ r is typable, then r mentions no unknowns.