to_ring_respects@{} : forall a b, equiv a b -> to_ring a = to_ring b. Proof. unfold equiv;intros [pa na] [pb nb] E. unfold to_ring;simpl in *. apply (left_cancellation (+) (naturals_to_semiring N B na + naturals_to_semiring N B nb)). path_via (naturals_to_semiring N B pa + naturals_to_semiring N B nb + 0); [rewrite <-(plus_negate_r (naturals_to_semiring N B na));ring_with_nat |rewrite plus_0_r]. path_via (naturals_to_semiring N B pb + naturals_to_semiring N B na + 0); [rewrite plus_0_r| rewrite <-(plus_negate_r (naturals_to_semiring N B nb));ring_with_nat]. rewrite <-2!preserves_plus. apply ap,E. Qed.
Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.