Computation and Deduction

ion and function application employ the notion of substitution. Recall the inference rule and its representation in LF: e1 ↪→ lam x. e1 e2 ↪→ v2 [v2/x]e1 ↪→ v ev app e1 e2 ↪→ v ev app : ΠE1:exp. ΠE2:exp. ΠE′ 1:exp→ exp. ΠV2:exp. ΠV :exp. eval E1 (lam E′ 1) → eval E2 V2 → eval (E′ 1 V2) V → eval (app E1 E2) V. As before, we transcribe this (and the trivial rule for evaluating λ-expressions) into Elf. ev_lam : eval (lam E) (lam E). ev_app : eval (app E1 E2) V exp will be instantiated to ([x:exp] !e1"). In the operational semantics of Elf, an application which is not in canonical form (such as (E1’ V2) after instantiation of E1’ and V2) will be reduced until it is in head-normal form (see Section ??)—in this case this means performing the substitution of V2 for the top-level bound variable in E1’. As an example, consider the evaluation of (lam x. x) z which is given by the deduction ev lam lam x. x ↪→ lam x. x ev z z ↪→ z ev z z ↪→ z ev app. (lam x. x) z ↪→ z