proposition: Lemma int_to_nat_of_nonpos x : x ≤ 0 → int_to_nat Z N x = 0.
proof: Proof. intros E. destruct (int_to_nat_spec x) as [[? E2]|]; intuition. apply int_to_nat_cancel_l. rewrite preserves_0. now apply (antisymmetry (≤)). Qed.
imports: Coq.setoid_ring.Ring MathClasses.interfaces.naturals MathClasses.interfaces.abstract_algebra MathClasses.interfaces.orders MathClasses.orders.nat_int MathClasses.theory.integers MathClasses.theory.rings MathClasses.orders.rings