We can make the following inferences.
Abstract:
As the example at the end of the previous section shows, unrestricted recursive types destroy the normalization property. This also means it is impossible to give all recursive types a logical interpretation. When we examine the inference rules we notice that recursive types are impredicative: the binder µα in µα. A ranges over the whole type. This means in the introduction rule, the type in the premiss [µα. A/α]A generally will be larger than the type µα. A in the conclusion. That alone is not responsible for non-termination: there are other type disciplines such as the polymorphic λ-calculus which retain a logical interpretation and termination, yet are impredicative. In this section we focus on the property that all well-typed terms in the linear λ-calculus without recursive types and fixpoint operators evaluate to a value. This is related to the normalization theorem for natural deductions (Theorem 3.10): if Γ; ∆ ⊢ A then Γ; ∆ ⊢ A ↑. We proved this by a rather circuitous route: unrestricted natural deductions can be translated to sequent derivations with cut from which we can eliminate cut and translate the result cut-free derivation back to a noraml natural deduction. Here, we prove directly that every term evaluates using the proof technique of logical relations [Sta85] also called Tait’s method [Tai67]. Because of the importance of this technique, we spend some time motivating its form. Our ultimate goal is to prove: If ·; ·⊢M:Athen M↩→V for some value V. The first natural attempt would be to prove this by induction on the typing derivation. Surprisingly, case for ⊸I works, even though we cannot apply the inductive hypothesis, since every linear λ-abstraction immediately evaluates to itself. Inthecasefor⊸E, however, we find that we cannot complete the proof. Let us examine why.
