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).Notation "[ x ]" := (phi x).Notation "x [=] y" := (ceqb x y).Notation "x [<=] y" := (cleb x y).(* Let's collect all hypotheses in addition to the ordered ring axioms into
one structure *)

Record SORaddon := mk_SOR_addon {
  SORrm : ring_morph 0 1 rplus rtimes rminus ropp req cO cI cplus ctimes cminus copp ceqb phi;
  SORpower : power_theory rI rtimes req pow_phi rpow;
  SORcneqb_morph : forall x y : C, x [=] y = false -> [x] ~= [y];
  SORcleb_morph : forall x y : C, x [<=] y = true -> [x] <= [y]
}.(* Let's collect all hypotheses in addition to the ordered ring axioms into
one structure *)

Record SORaddon := mk_SOR_addon {
  SORrm : ring_morph 0 1 rplus rtimes rminus ropp req cO cI cplus ctimes cminus copp ceqb phi;
  SORpower : power_theory rI rtimes req pow_phi rpow;
  SORcneqb_morph : forall x y : C, x [=] y = false -> [x] ~= [y];
  SORcleb_morph : forall x y : C, x [<=] y = true -> [x] <= [y]
}.(* Let's collect all hypotheses in addition to the ordered ring axioms into
one structure *)

Record SORaddon := mk_SOR_addon {
  SORrm : ring_morph 0 1 rplus rtimes rminus ropp req cO cI cplus ctimes cminus copp ceqb phi;
  SORpower : power_theory rI rtimes req pow_phi rpow;
  SORcneqb_morph : forall x y : C, x [=] y = false -> [x] ~= [y];
  SORcleb_morph : forall x y : C, x [<=] y = true -> [x] <= [y]
}.(* Let's collect all hypotheses in addition to the ordered ring axioms into
one structure *)

Record SORaddon := mk_SOR_addon {
  SORrm : ring_morph 0 1 rplus rtimes rminus ropp req cO cI cplus ctimes cminus copp ceqb phi;
  SORpower : power_theory rI rtimes req pow_phi rpow;
  SORcneqb_morph : forall x y : C, x [=] y = false -> [x] ~= [y];
  SORcleb_morph : forall x y : C, x [<=] y = true -> [x] <= [y]
}.(* Let's collect all hypotheses in addition to the ordered ring axioms into
one structure *)

Record SORaddon := mk_SOR_addon {
  SORrm : ring_morph 0 1 rplus rtimes rminus ropp req cO cI cplus ctimes cminus copp ceqb phi;
  SORpower : power_theory rI rtimes req pow_phi rpow;
  SORcneqb_morph : forall x y : C, x [=] y = false -> [x] ~= [y];
  SORcleb_morph : forall x y : C, x [<=] y = true -> [x] <= [y]
}.(* Let's collect all hypotheses in addition to the ordered ring axioms into
one structure *)

Record SORaddon := mk_SOR_addon {
  SORrm : ring_morph 0 1 rplus rtimes rminus ropp req cO cI cplus ctimes cminus copp ceqb phi;
  SORpower : power_theory rI rtimes req pow_phi rpow;
  SORcneqb_morph : forall x y : C, x [=] y = false -> [x] ~= [y];
  SORcleb_morph : forall x y : C, x [<=] y = true -> [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 micomega_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 Morphism rminus with signature req ==> req ==> req as rminus_morph.Definition cneqb (x y : C) := negb (ceqb x y).Definition cltb (x y : C) := (cleb x y) && (cneqb x y).Notation "x [~=] y" := (cneqb x y).Notation "x [<] y" := (cltb x y).Ltac le_less := rewrite (Rle_lt_eq sor); left; try assumption.Ltac le_equal := rewrite (Rle_lt_eq sor); right; try reflexivity; try assumption.Ltac le_elim H := rewrite (Rle_lt_eq sor) in H; destruct H as [H | H].

(* Specifications: *)
Lemma cleb_sound : forall x y : C, x [<=] y = true -> [x] <= [y].
Lemma cneqb_sound : forall x y : C, x [~=] y = true -> [x] ~= [y].
Lemma cltb_sound : forall x y : C, x [<] y = true -> [x] < [y].