Object Oriented Design

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By Harrison J.

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If t is a combination s u then we have for some τ that Γ s : τ → σ and Γ u : τ . By the inductive hypothesis, we have Γs s : τ → σ and Γu u : τ . By the monotonicity lemma, we therefore have Γsu s : τ → σ and Γsu u : τ , since F V (s u) = F V (s) ∪ F V (u). Hence by the rule for combinations we get Γsu t : σ. Finally, if t has the form λx. s, we must have 44 CHAPTER 4. TYPES Γ ∪ {x : τ } s : τ with σ of the form τ → τ . By the inductive hypothesis we have (Γ ∪ {x : τ })s s : τ , and so (Γ ∪ {x : τ })s − {x : τ } (λx.

N )con for the application of an n-ary type constructor con to the arguments αi . 1. ) For example (σ)list is the type of lists whose elements all have type σ. An important property of the types, provable because of free inductive generation, is that σ → τ = σ. ) This rules out the possibility of applying a term to itself, unless the two instances in question have distinct types. 2 Church and Curry typing There are two major approaches to defining typed lambda calculus. One approach, due to Church, is explicit.

En )) We need simply say that the infix comma operator is right-associative, and can understand this without any other conventions. For example: (p, q, r, s) = = = = = (p, (q, (r, s))) λf. f p (q, (r, s)) λf. f p (λf. f q (r, s)) λf. f p (λf. f q (λf. f r s)) λf. f p (λg. g q (λh. h r s)) where in the last line we have performed an alpha-conversion for the sake of clarity. Although tuples are built up in a flat manner, it is easy to create arbitrary finitely-branching tree structures by using tupling repeatedly.

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