Section STRICT_ORDERED_RING.
Variable R : Type.
Variable (rO rI : R) (rplus rtimes rminus: R -> R -> R) (ropp : R -> R).
Variable req rle rlt : R -> R -> Prop.
Variable sor : SOR rO rI rplus rtimes rminus ropp req rle rlt.
Notation "0" := rO.
Notation "1" := rI.
Notation "x + y" := (rplus x y).
Notation "x * y " := (rtimes x y).
Notation "x - y " := (rminus x y).
Notation "- x" := (ropp x).
Notation "x == y" := (req x y).
Notation "x ~= y" := (~ req x y).
Notation "x <= y" := (rle x y).
Notation "x < y" := (rlt x y).
Add Relation R req
reflexivity proved by (@Equivalence_Reflexive _ _ (SORsetoid sor))
symmetry proved by (@Equivalence_Symmetric _ _ (SORsetoid sor))
transitivity proved by (@Equivalence_Transitive _ _ (SORsetoid sor))
as sor_setoid.
Add Morphism rplus with signature req ==> req ==> req as rplus_morph.
Add Morphism rtimes with signature req ==> req ==> req as rtimes_morph.
Add Morphism ropp with signature req ==> req as ropp_morph.
Add Morphism rle with signature req ==> req ==> iff as rle_morph.
Add Morphism rlt with signature req ==> req ==> iff as rlt_morph.
Add Ring SOR : (SORrt sor).
Add Morphism rminus with signature req ==> req ==> req as rminus_morph.
(* Properties of plus, minus and opp *)
(* Relations *)
Proof (SORle_refl sor).
Proof (SORle_antisymm sor).
Proof (SORle_trans sor).
Proof (SORlt_trichotomy sor).
Proof (SORlt_le_neq sor).
Proof (SORneq_0_1 sor).
Ltac le_less := rewrite Rle_lt_eq; left; try assumption.
Ltac le_equal := rewrite Rle_lt_eq; right; try reflexivity; try assumption.
Ltac le_elim H := rewrite Rle_lt_eq in H; destruct H as [H | H].
(* Plus, minus and order *)

Theorem Rplus_le_lt_mono : forall n m p q : R, n <= m -> p < q -> n + p < m + q.