Factorial

Mathematical statement

The function F:N→N is uniquely determined recursively by F(0)=1 and F(n+1)=(n+1)F(n). Its customary notation is n!.

Representative formula (exact TeX)

F(n)

Assumptions

• n is a natural number, with natural numbers including 0.

Symbol scope

• F — representative factorial function
• n — natural-number input
• ! — customary postfix factorial notation

Intuition, definition, and misconceptions

The value 0! is defined as 1. For a natural number n, (n+1)!=(n+1)·n!, so when n>0, n!=1·2·…·n.

Factorial grows rapidly. Exact computation should use arbitrary-precision integers rather than presenting a floating-point approximation as an exact value.

Factorial occurs in permutation counting, but “n! counts the permutations of an n-element set” is a separate theorem requiring proof; it is not the recursive definition itself.

Common mistakes include treating 0! as 0, reading n! as n multiplied by itself, or presenting an unsafe floating-point value as an exact result.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Natural number — Factorial inputs and outputs are natural numbers.
• Arithmetic operations — The successor recurrence and finite product use natural-number multiplication.

Verification boundary

BigInt can recompute factorial values and recurrence witnesses for finite natural inputs, but does not prove permutation-counting, divisibility, or asymptotic theorems.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind Nat.factorial, factorial_zero, factorial_succ, and this MKO’s natural-number convention.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.