Remarks on Krivine's " Lambda-calculus, Types and Models " , Chapter 1, §2

The point of this note is to 1) add some lemmata to Chapter 1 §2 of [1] (lemmata that are used in [1] without mention, due to their intuitive obviousness); 2) show that the definition of α-equivalence given in [1] is equivalent to the definition of α-equivalence given in some other sources; 3) prove some rules for substitution (in order to answer a MathOverflow question of myself). We are going to use the notations and the results of Chapter 1 of [1]. In particular, the sign ≡ will stand for the α-equivalence defined in [1]. The different notion of α-equivalence that we consider will be denoted by = α (in order not to confuse it with ≡ as long as it is not yet proven that the two notions are equivalent). Here come several facts silently used in some proofs in §1.2 of [1]. These facts are all pretty simple, intuitively clear and easy to prove, and I suspect this is why they have not been explicitly stated in [1]. I am making them explicit and proving them in detail in order to formalize the theory a little bit more. We begin with some properties of bound variables (and their behaviour under substitution). Definition: If u is a term in L, let BV u denote the set of bounded variables of the term u. Before we continue, let us give an inductive method to compute BV u for a term u: If u = x for a variable x, then BV u = ∅. If u = (v) w for terms v and w, then BV u = (BV v) ∪ (BV w). If u = λxv for some variable x and some term v, then BV u = {x} ∪ (BV v). Proof of Lemma 1.A. We proceed by induction over t: If t is a variable or a term of the form (u) v, the induction step is clear. Remains to consider the case when t = λxu for some variable x and some term u. In this case, BV t = {x} ∪ (BV u). There are two subcases to consider: the subcase when x ∈ {x 1 , .