(** Reflexivity **)
  Inductive makeRefl (x : T) : T -> Prop :=
  | RRefl : makeRefl x x
  | RStep : forall y, R x y -> makeRefl x y.(** Reflexivity **)
  Inductive makeRefl (x : T) : T -> Prop :=
  | RRefl : makeRefl x x
  | RStep : forall y, R x y -> makeRefl x y.(** Reflexivity **)
  Inductive makeRefl (x : T) : T -> Prop :=
  | RRefl : makeRefl x x
  | RStep : forall y, R x y -> makeRefl x y.Global Instance Refl_makeRefl : Reflexive makeRefl.Global Instance Refl_makeTrans : Transitive R -> Transitive makeRefl.(** Transitivity **)
  Inductive makeTrans (x y : T) : Prop :=
  | TStep : R x y -> makeTrans x y
  | TTrans : forall z, makeTrans x z -> makeTrans z y -> makeTrans x y.(** Transitivity **)
  Inductive makeTrans (x y : T) : Prop :=
  | TStep : R x y -> makeTrans x y
  | TTrans : forall z, makeTrans x z -> makeTrans z y -> makeTrans x y.(** Transitivity **)
  Inductive makeTrans (x y : T) : Prop :=
  | TStep : R x y -> makeTrans x y
  | TTrans : forall z, makeTrans x z -> makeTrans z y -> makeTrans x y.Global Instance Trans_makeTrans : Transitive makeTrans.Global Instance Trans_makeRefl : Reflexive R -> Reflexive makeTrans.Inductive leftTrans (x y : T) : Prop :=
  | LTFin : R x y -> leftTrans x y
  | LTStep : forall z, R x z -> leftTrans z y -> leftTrans x y.Inductive leftTrans (x y : T) : Prop :=
  | LTFin : R x y -> leftTrans x y
  | LTStep : forall z, R x z -> leftTrans z y -> leftTrans x y.Inductive leftTrans (x y : T) : Prop :=
  | LTFin : R x y -> leftTrans x y
  | LTStep : forall z, R x z -> leftTrans z y -> leftTrans x y.Inductive rightTrans (x y : T) : Prop :=
  | RTFin : R x y -> rightTrans x y
  | RTStep : forall z, rightTrans x z -> R z y -> rightTrans x y.Inductive rightTrans (x y : T) : Prop :=
  | RTFin : R x y -> rightTrans x y
  | RTStep : forall z, rightTrans x z -> R z y -> rightTrans x y.Inductive rightTrans (x y : T) : Prop :=
  | RTFin : R x y -> rightTrans x y
  | RTStep : forall z, rightTrans x z -> R z y -> rightTrans x y.(** Equivalence of definitions of transitivity **)
  Fixpoint leftTrans_rightTrans_acc x y (l : leftTrans y x) : forall z, rightTrans z y -> rightTrans z x :=
    match l with
      | LTFin pf => fun z pfR => RTStep pfR pf
      | LTStep _ pf pfL => fun z pfR =>
        leftTrans_rightTrans_acc pfL (RTStep pfR pf)
    end.Fixpoint rightTrans_leftTrans_acc x y (l : rightTrans x y) : forall z, leftTrans y z -> leftTrans x z :=
    match l with
      | RTFin pf => fun z pfR => LTStep pf pfR
      | RTStep _ pf pfL => fun z pfR =>
        rightTrans_leftTrans_acc pf (LTStep pfL pfR)
    end.

(* Specifications: *)
Theorem leftTrans_rightTrans : forall x y, leftTrans x y <-> rightTrans x y.