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By Barendregt H., Barendsen E.
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What is the most general type for this term? 2. xx) have no type in λ→. 3. Find the most general types (if they exist) for the following terms. xyy. (ii) SII. z(yx)). 4. Find terms M, N ∈ Λ such that the following hold in λ→. (i) M : (α→β)→(β→γ)→(α→γ). (ii) N : (((α→β)→β)→β)→(α→β). 5. 3. Chapter 6 Extensions In Chapter 3 we have seen that all computable functions can be expressed in the lambda calculus. For reasons of efficiency, reliability and convenience this language will be extended. The set of λ-terms Λ will be extended with constants.
I) If Γ ⊇ Γ is another basis, then Γ (ii) Γ (iii) Γ M :σ ⇒ Γ M : σ. M : σ ⇒ FV(M ) ⊆ dom(Γ). M : σ ⇒ Γ FV(M ) M : σ. Proof. (i) By induction on the derivation of M : σ. Since such proofs will occur frequently we will spell it out in this simple situation in order to be shorter later on. Type Assignment 37 Case 1. M : σ is x:σ and is element of Γ. Then also x:σ ∈ Γ and hence Γ M : σ. Case 2. M : σ is (M1 M2 ) : σ and follows directly from M1 : (τ →σ) and M2 : τ for some τ . By the IH one has Γ M1 : (τ →σ) and Γ M2 : τ .
An unsolvable problem of elementary number theory, American Journal of Mathematics 58, pp. 354–363. Church, A. (1940). A formulation of the simple theory of types, Journal of Symbolic Logic 5, pp. 56–68. Church, A. (1941). The Theory of Lambda Conversion, Princeton University Press. B. (1934). Functionality in combinatory logic, Proceedings of the National Academy of Science USA 20, pp. 584–590. B. (1969). Modified basic functionality in combinatory logic, Dialectica 23, pp. 83–92. B. and R. Feys (1958).