#[export] Instance Rgt_Transitive : Transitive Rgt | 10 := Rgt_trans.

(* Specifications: *)
Lemma Rle_lt_trans : forall r1 r2 r3, r1 <= r2 -> r2 < r3 -> r1 < r3.
Lemma Rlt_le_trans : forall r1 r2 r3, r1 < r2 -> r2 <= r3 -> r1 < r3.
Lemma Rge_gt_trans : forall r1 r2 r3, r1 >= r2 -> r2 > r3 -> r1 > r3.
Lemma Rgt_ge_trans : forall r1 r2 r3, r1 > r2 -> r2 >= r3 -> r1 > r3.
Lemma Rlt_dec : forall r1 r2, {r1 < r2} + {~ r1 < r2}.
Lemma Rle_dec : forall r1 r2, {r1 <= r2} + {~ r1 <= r2}.
Lemma Rgt_dec : forall r1 r2, {r1 > r2} + {~ r1 > r2}.
Lemma Rge_dec : forall r1 r2, {r1 >= r2} + {~ r1 >= r2}.
Lemma Rlt_le_dec : forall r1 r2, {r1 < r2} + {r2 <= r1}.
Lemma Rlt_ge_dec : forall r1 r2, {r1 < r2} + {r1 >= r2}.
Lemma Rgt_ge_dec : forall r1 r2, {r1 > r2} + {r2 >= r1}.
Lemma Rgt_le_dec : forall r1 r2, {r1 > r2} + {r1 <= r2}.
Lemma Rle_lt_dec : forall r1 r2, {r1 <= r2} + {r2 < r1}.
Lemma Rle_gt_dec : forall r1 r2, {r1 <= r2} + {r1 > r2}.
Lemma Rge_gt_dec : forall r1 r2, {r1 >= r2} + {r2 > r1}.
Lemma Rge_lt_dec : forall r1 r2, {r1 >= r2} + {r1 < r2}.
Lemma Rle_lt_or_eq_dec : forall r1 r2, r1 <= r2 -> {r1 < r2} + {r1 = r2}.
Lemma Rge_gt_or_eq_dec : forall r1 r2, r1 >= r2 -> {r1 > r2} + {r1 = r2}.
Lemma Private_sumbool_to_or {A B : Prop} : {A} + {B} -> A \/ B.
Lemma Rlt_or_not_lt : forall r1 r2, r1 < r2 \/ ~(r1 < r2).
Lemma Rle_or_not_le : forall r1 r2, r1 <= r2 \/ ~(r1 <= r2).
Lemma Rgt_or_not_gt : forall r1 r2, r1 > r2 \/ ~(r1 > r2).
Lemma Rge_or_not_ge : forall r1 r2, r1 >= r2 \/ ~(r1 >= r2).
Lemma Rlt_or_le : forall r1 r2, r1 < r2 \/ r2 <= r1.
Lemma Rgt_or_ge : forall r1 r2, r1 > r2 \/ r2 >= r1.
Lemma Rle_or_lt : forall r1 r2, r1 <= r2 \/ r2 < r1.
Lemma Rge_or_gt : forall r1 r2, r1 >= r2 \/ r2 > r1.
Lemma Rlt_or_ge : forall r1 r2, r1 < r2 \/ r1 >= r2.
Lemma Rgt_or_le : forall r1 r2, r1 > r2 \/ r1 <= r2.
Lemma Rle_or_gt : forall r1 r2, r1 <= r2 \/ r1 > r2.
Lemma Rge_or_lt : forall r1 r2, r1 >= r2 \/ r1 < r2.
Lemma Rle_lt_or_eq : forall r1 r2, r1 <= r2 -> r1 < r2 \/ r1 = r2.
Lemma Rge_gt_or_eq : forall r1 r2, r1 >= r2 -> r1 > r2 \/ r1 = r2.
Lemma Rplus_eq_compat_l : forall r r1 r2, r1 = r2 -> r + r1 = r + r2.
Lemma Rplus_eq_compat_r : forall r r1 r2, r1 = r2 -> r1 + r = r2 + r.
Lemma Rplus_0_r : forall r, r + 0 = r.
Lemma Rplus_ne : forall r, r + 0 = r /\ 0 + r = r.
Lemma Rplus_opp_l : forall r, - r + r = 0.
Lemma Rplus_opp_r_uniq : forall r1 r2, r1 + r2 = 0 -> r2 = - r1.