# De Bruijn index

## Mathematical notation in lambda calculus / From Wikipedia, the free encyclopedia

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In mathematical logic, the **de Bruijn index** is a tool invented by the Dutch mathematician Nicolaas Govert de Bruijn for representing terms of lambda calculus without naming the bound variables.[1] Terms written using these indices are invariant with respect to α-conversion, so the check for α-equivalence is the same as that for syntactic equality. Each de Bruijn index is a natural number that represents an occurrence of a variable in a λ-term, and denotes the number of binders that are in scope between that occurrence and its corresponding binder. The following are some examples:

- The term λ
*x*. λ*y*.*x*, sometimes called the K combinator, is written as λ λ 2 with de Bruijn indices. The binder for the occurrence*x*is the second λ in scope. - The term λ
*x*. λ*y*. λ*z*.*x**z*(*y**z*) (the S combinator), with de Bruijn indices, is λ λ λ 3 1 (2 1). - The term λ
*z*. (λ*y*.*y*(λ*x*.*x*)) (λ*x*.*z**x*) is λ (λ 1 (λ 1)) (λ 2 1). See the following illustration, where the binders are colored and the references are shown with arrows.

De Bruijn indices are commonly used in higher-order reasoning systems such as automated theorem provers and logic programming systems.[2]

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