By Jean-Louis Krivine, Rene Corvi (translator)

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**Example text**

Tn } (this is indeed a saturated subset of Λ). It follows from this deﬁnition that x ∈ |B1 , . . , Bn → X|I . Thus : x ∈ | ni=1 Ai ∧ (B1 , . . , Bn → X)|I . Let v ∈ |X|I , with no free variables but x1 , . . , xk . Then v reduces to (x)t1 . . tn by leftmost β-reduction ; we have ti ∈ |Bi |I and therefore, by induction hypothesis, ti ti , where ti is an η-reduced image of ti . Hence v (x)t1 . . tn , which is clearly an η-reduced image of t = (x)t1 . . tn . So we have shown that the interpretation I satisﬁes all the required properties with respect to the given principal typing of t.

As an immediate consequence, we have : If A is a trivial type, then its value |A|I under any interpretation I is the whole set Λ. 6. Let N0 , N be subsets of Λ, with the following properties : N is saturated, N0 ⊂ N , N0 ⊂ (Λ → N0 ), N ⊃ (N0 → N ). Let I be the interpretation such that |X|I = N for every type variable X. Then |A|I ⊃ N0 for every type A, and |A|I ⊂ N for every non-trivial type A. We ﬁrst prove, by induction on A, that |A|I ⊃ N0 ; this is obvious whenever A is a type variable, or A = Ω, or A = B ∧ C.

Yl (y)v1 . . vp be another head normal form of t. 24, there exists a term t2 which can be obtained by βreduction from t0 as well as from t1 . Now, in t0 (resp. t1 ) all possible β-reductions have to be made in u1 , . . , un (resp. v1 , . . , vp ). Hence : t2 ≡ λx1 . . λxk (x)u1 . . un ≡ λy1 . . λyl (y)v1 . . vp with ui β ui , vj β vj . This yields the expected result. 3. For every λ-term t, the following conditions are equivalent : i) t is solvable ; ii) t is β-equivalent to a head normal form ; iii) the head reduction of t terminates (with a head normal form).