proposition: Lemma int_to_nat_le_back x y : 0 ≤ y → int_to_nat Z N x ≤ int_to_nat Z N y → x ≤ y.
proof: Proof. intros. rewrite <-(int_to_nat_of_nonneg y) by easy. transitivity (f (int_to_nat Z N x)). now apply int_to_nat_le_r. now apply (order_preserving f). 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