prime_two
plain-language theorem explainer
The declaration asserts that 2 satisfies the primality predicate on natural numbers. Number theorists constructing modular sieves or wheel factorizations inside the Recognition monolith cite it as the initial case for small prime divisors. The proof reduces to a single decision procedure that evaluates the decidable primality check directly.
Claim. The natural number 2 is prime, i.e., it satisfies the predicate that a natural number is prime.
background
The module supplies basic prime-number footholds for the reality repository while reusing Mathlib's Nat.Prime. Prime(n) is defined locally as an abbreviation for Nat.Prime n and kept transparent so that Mathlib lemmas remain directly applicable. The surrounding sanity theorems, including prime_three and not_prime_one, confirm that the namespace is axiom-free and correctly wired before any ascent to analytic number theory.
proof idea
The proof is a one-line wrapper that invokes the decide tactic on the decidable proposition Nat.Prime 2.
why it matters
This result supplies the base case for theorems that classify prime divisors of wheel840, such as prime_dvd_wheel840 which concludes any prime divisor must be 2, 3, 5 or 7. It likewise supports wheel840_accepts by identifying the small primes that divide the wheel. Within the Recognition framework it provides an axiom-free foothold for number-theoretic constructions that feed into larger modular arguments.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.