Infix "<d" := divides (at level 70, no associativity).Definition prime p := p <> 1 /\ forall q, q <d p -> q = 1 \/ q = p.Fact prime_2 : prime 2.Fact prime_ge_2 p : prime p -> 2 <= p.Fact prime_gcd p q : prime p -> is_gcd p q 1 \/ p <d q.Fact prime_div_mult p x y : prime p -> p <d x*y -> p <d x \/ p <d y.Definition prime_or_div p : 2 <= p -> { q | 2 <= q < p /\ q <d p } + { prime p }.

(* Specifications: *)
Theorem prime_factor n : 2 <= n -> { p | prime p /\ p <d n }.
Theorem prime_rect n : P n.