Module Import MO := OrderedTypeFacts O.
Local Open Scope order.

Lemma lt_not_eq x y : x<y -> ~x==y.  Proof. order. Qed.
Lemma lt_eq x y z : x<y -> y==z -> x<z. Proof. order. Qed.
Lemma eq_lt x y z : x==y -> y<z -> x<z. Proof. order. Qed.
Lemma le_eq x y z : x<=y -> y==z -> x<=z. Proof. order. Qed.
Lemma eq_le x y z : x==y -> y<=z -> x<=z. Proof. order. Qed.
Lemma neq_eq x y z : ~x==y -> y==z -> ~x==z. Proof. order. Qed.
Lemma eq_neq x y z : x==y -> ~y==z -> ~x==z. Proof. order. Qed.
Lemma le_lt_trans x y z : x<=y -> y<z -> x<z. Proof. order. Qed.
Lemma lt_le_trans x y z : x<y -> y<=z -> x<z. Proof. order. Qed.
Lemma le_trans x y z : x<=y -> y<=z -> x<=z. Proof. order. Qed.
Lemma le_antisym x y : x<=y -> y<=x -> x==y. Proof. order. Qed.
Lemma le_neq x y : x<=y -> ~x==y -> x<y. Proof. order. Qed.
Lemma neq_sym x y : ~x==y -> ~y==x. Proof. order. Qed.
Lemma lt_le x y : x<y -> x<=y. Proof. order. Qed.
Lemma gt_not_eq x y : y<x -> ~x==y. Proof. order. Qed.
Lemma eq_not_lt x y : x==y -> ~x<y. Proof. order. Qed.
Lemma eq_not_gt x y : x==y -> ~ y<x. Proof. order. Qed.
Lemma lt_not_gt x y : x<y -> ~ y<x. Proof. order. Qed.
Lemma eq_is_nlt_ngt x y : x==y <-> ~x<y /\ ~y<x.