#[export] Instance Rgt_Irreflexive : Irreflexive Rgt | 10 := Rgt_irrefl.

(* Specifications: *)
Lemma Rlt_not_eq : forall r1 r2, r1 < r2 -> r1 <> r2.
Lemma Rgt_not_eq : forall r1 r2, r1 > r2 -> r1 <> r2.
Lemma Rlt_dichotomy_converse : forall r1 r2, r1 < r2 \/ r1 > r2 -> r1 <> r2.
Lemma Rtotal_order : forall r1 r2, r1 < r2 \/ r1 = r2 \/ r1 > r2.
Lemma Req_dec : forall r1 r2:R, r1 = r2 \/ r1 <> r2.
Lemma Rdichotomy : forall r1 r2, r1 <> r2 -> r1 < r2 \/ r1 > r2.
Lemma Req_dec_T : forall r1 r2:R, {r1 = r2} + {r1 <> r2}.
Lemma Rlt_le : forall r1 r2, r1 < r2 -> r1 <= r2.
Lemma Rgt_ge : forall r1 r2, r1 > r2 -> r1 >= r2.
Lemma Rle_ge : forall r1 r2, r1 <= r2 -> r2 >= r1.
Lemma Rge_le : forall r1 r2, r1 >= r2 -> r2 <= r1.
Lemma Rlt_gt : forall r1 r2, r1 < r2 -> r2 > r1.
Lemma Rgt_lt : forall r1 r2, r1 > r2 -> r2 < r1.
Lemma Rnot_le_lt : forall r1 r2, ~ r1 <= r2 -> r2 < r1.
Lemma Rnot_ge_gt : forall r1 r2, ~ r1 >= r2 -> r2 > r1.
Lemma Rnot_le_gt : forall r1 r2, ~ r1 <= r2 -> r1 > r2.
Lemma Rnot_ge_lt : forall r1 r2, ~ r1 >= r2 -> r1 < r2.
Lemma Rnot_lt_le : forall r1 r2, ~ r1 < r2 -> r2 <= r1.
Lemma Rnot_gt_le : forall r1 r2, ~ r1 > r2 -> r1 <= r2.
Lemma Rnot_gt_ge : forall r1 r2, ~ r1 > r2 -> r2 >= r1.
Lemma Rnot_lt_ge : forall r1 r2, ~ r1 < r2 -> r1 >= r2.
Lemma Rlt_not_le : forall r1 r2, r2 < r1 -> ~ r1 <= r2.
Lemma Rgt_not_le : forall r1 r2, r1 > r2 -> ~ r1 <= r2.
Lemma Rlt_not_ge : forall r1 r2, r1 < r2 -> ~ r1 >= r2.
Lemma Rgt_not_ge : forall r1 r2, r2 > r1 -> ~ r1 >= r2.
Lemma Rle_not_lt : forall r1 r2, r2 <= r1 -> ~ r1 < r2.
Lemma Rge_not_lt : forall r1 r2, r1 >= r2 -> ~ r1 < r2.
Lemma Rle_not_gt : forall r1 r2, r1 <= r2 -> ~ r1 > r2.
Lemma Rge_not_gt : forall r1 r2, r2 >= r1 -> ~ r1 > r2.
Lemma Req_le_sym : forall r1 r2, r2 = r1 -> r1 <= r2.
Lemma Req_ge_sym : forall r1 r2, r2 = r1 -> r1 >= r2.