# Notation ℕ for the natural numbers. #

The type of natural numbers, starting at zero. It is defined as an inductive type freely generated by "zero is a natural number" and "the successor of a natural number is a natural number".

You can prove a theorem P n about n : Nat by induction n, which will expect a proof of the theorem for P 0, and a proof of P (succ i) assuming a proof of P i. The same method also works to define functions by recursion on natural numbers: induction and recursion are two expressions of the same operation from Lean's point of view.

open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih


This type is special-cased by both the kernel and the compiler:

• The type of expressions contains "Nat literals" as a primitive constructor, and the kernel knows how to reduce zero/succ expressions to nat literals.
• If implemented naively, this type would represent a numeral n in unary as a linked list with n links, which is horribly inefficient. Instead, the runtime itself has a special representation for Nat which stores numbers up to 2^63 directly and larger numbers use an arbitrary precision "bignum" library (usually GMP).
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