(**  Morphism property for unary and binary functions  **)
Definition morphism_1 aF rF := forall x, f (aF x) = rF (f x).Definition morphism_2 aOp rOp := forall x y, f (aOp x y) = rOp (f x) (f y).(**  Homomorphism property for unary and binary relations  **)
Definition homomorphism_1 (aP rP : _ -> Prop) := forall x, aP x -> rP (f x).Definition homomorphism_2 (aR rR : _ -> _ -> Prop) :=
  forall x y, aR x y -> rR (f x) (f y).(**  Stability property for unary and binary relations  **)
Definition monomorphism_1 (aP rP : _ -> sT) := forall x, rP (f x) = aP x.Definition monomorphism_2 (aR rR : _ -> _ -> sT) :=
  forall x y, rR (f x) (f y) = aR x y.Notation "{ 'morph' f : x / a >-> r }" :=
  (morphism_1 f (fun x => a) (fun x => r)) : type_scope.Notation "{ 'morph' f : x / a }" :=
  (morphism_1 f (fun x => a) (fun x => a)) : type_scope.Notation "{ 'morph' f : x y / a >-> r }" :=
  (morphism_2 f (fun x y => a) (fun x y => r)) : type_scope.Notation "{ 'morph' f : x y / a }" :=
  (morphism_2 f (fun x y => a) (fun x y => a)) : type_scope.Notation "{ 'homo' f : x / a >-> r }" :=
  (homomorphism_1 f (fun x => a) (fun x => r)) : type_scope.Notation "{ 'homo' f : x / a }" :=
  (homomorphism_1 f (fun x => a) (fun x => a)) : type_scope.Notation "{ 'homo' f : x y / a >-> r }" :=
  (homomorphism_2 f (fun x y => a) (fun x y => r)) : type_scope.Notation "{ 'homo' f : x y / a }" :=
  (homomorphism_2 f (fun x y => a) (fun x y => a)) : type_scope.Notation "{ 'homo' f : x y /~ a }" :=
  (homomorphism_2 f (fun y x => a) (fun x y => a)) : type_scope.Notation "{ 'mono' f : x / a >-> r }" :=
  (monomorphism_1 f (fun x => a) (fun x => r)) : type_scope.Notation "{ 'mono' f : x / a }" :=
  (monomorphism_1 f (fun x => a) (fun x => a)) : type_scope.Notation "{ 'mono' f : x y / a >-> r }" :=
  (monomorphism_2 f (fun x y => a) (fun x y => r)) : type_scope.Notation "{ 'mono' f : x y / a }" :=
  (monomorphism_2 f (fun x y => a) (fun x y => a)) : type_scope.Notation "{ 'mono' f : x y /~ a }" :=
  (monomorphism_2 f (fun y x => a) (fun x y => a)) : type_scope.Definition injective := forall x1 x2, f x1 = f x2 -> x1 = x2.Definition cancel g := forall x, g (f x) = x.Definition pcancel g := forall x, g (f x) = Some x.Definition ocancel (g : aT -> option rT) h := forall x, oapp h x (g x) = x.

Lemma etrans_id T x y (eqxy : x = y :> T) : etrans (erefl x) eqxy = eqxy.