By Jean-Louis Krivine, Rene Corvi (translator)

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Uk is of type X in the context y : U (where U = Ω, Ω, . . , Ω → X). Thus t is of type U1 , . . , Un → X in the context y : U (U1 , . . , Un may be arbitrarily chosen, except when y = xi ; in that case, take Ui = U). D. 22. If x1 : A1 , x2 : A2 , . . , xk : Ak x1 : A1 ∧ A1 , x2 : A2 , . . , xk : Ak t : A. t : A, then : Proof by induction on the number of rules used to obtain : t : A (either rules 1 to 6, page 42 or x1 : A1 , x2 : A2 ,. . , xk : Ak rules 1 to 5, page 51). Consider the last one.

Of terms such that : for every n ≥ 0, tn β0 tn+1 (tn+1 is obtained by reducing a redex in tn ) ; for every n ≥ 0, there exists a p ≥ n such that tp+1 is obtained by reducing the leftmost redex in tp . 12 (Quasi leftmost normalization theorem). Suppose x1 : A1 , . . , xk : Ak DΩ t : A, and Ω does not occur in A,A1 ,. . ,Ak . Then there is no infinite quasi leftmost reduction of t. Chapter 3. Intersection type systems 49 In order to prove it, we again deﬁne an adapted pair (N0 , N ) : N is the set of all terms which do not admit an inﬁnite quasi leftmost reduction ; N0 is the set of all terms of the form (x)t1 .

In particular, all variables are in N0 (take n = 0). We have to check conditions (i) and (ii) in the deﬁnition of adapted pairs (page 46) : i) N is saturated : clearly, if (u[t/x])t1 . . tn is normalizable by leftmost β-reduction, then so is (λx u)tt1 . . tn . ii) N0 ⊂ N : if t ∈ N0 , say t = (x)t1 . . tn for some variable x and t1 , . . , tn ∈ N , then t1 , . . , tn are all normalizable by leftmost β-reduction, thus t clearly satisﬁes the same property. The inclusion N0 ⊂ (N → N0 ) is obvious.