De Bruijn index

In mathematical logic, the De Bruijn index is a notation invented by the Dutch mathematician Nicolaas Govert de Bruijn for representing terms in the λ calculus with the purpose of eliminating the names of the variable from the notation. 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:

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

Formally, λ-terms (M, N, …) written using De Bruijn indices have the following syntax (parentheses allowed freely):

where n—natural numbers greater than 0—are the variables. A variable n is bound if it is in the scope of at least n binders (λ); otherwise it is free. The binding site for a variable n is the nth binder it is in the scope of, starting from the innermost binder.

The most primitive operation on λ-terms is substitution: replacing free variables in a term with other terms. In the β-reduction (λ M) N, for example, we must:

...
Wikipedia