proposition: Lemma Q_sum_eq_join_meet@{} : forall a b c d : Q, a + b = c + d -> a + b = join a c + meet b d.
proof: Proof. intros ???? E. destruct (total le a c) as [E1|E1]. - rewrite (join_r _ _ E1). rewrite meet_r;trivial. apply (order_preserving (+ b)) in E1. rewrite E in E1. apply (order_reflecting (c +)). trivial. - rewrite (join_l _ _ E1). rewrite meet_l;trivial. apply (order_reflecting (a +)). rewrite E. apply (order_preserving (+ d)). trivial. Qed.
imports: HoTT.Classes.implementations.peano_naturals HoTT.Classes.implementations.natpair_integers HoTT.Classes.interfaces.abstract_algebra HoTT.Classes.interfaces.naturals HoTT.Classes.interfaces.rationals HoTT.Classes.interfaces.orders HoTT.Classes.theory.groups HoTT.Classes.theory.integers HoTT.Classes.theory.dec_fields HoTT.Classes.orders.sum HoTT.Classes.orders.dec_fields HoTT.Classes.orders.lattices HoTT.Classes.theory.additional_operations HoTT.Classes.tactics.ring_quote HoTT.Classes.tactics.ring_tac