(** Definition by direct transitive closure *)

  Inductive clos_trans (x: A) : A -> Prop :=
    | t_step (y:A) : R x y -> clos_trans x y
    | t_trans (y z:A) : clos_trans x y -> clos_trans y z -> clos_trans x z.(** Definition by direct transitive closure *)

  Inductive clos_trans (x: A) : A -> Prop :=
    | t_step (y:A) : R x y -> clos_trans x y
    | t_trans (y z:A) : clos_trans x y -> clos_trans y z -> clos_trans x z.(** Definition by direct transitive closure *)

  Inductive clos_trans (x: A) : A -> Prop :=
    | t_step (y:A) : R x y -> clos_trans x y
    | t_trans (y z:A) : clos_trans x y -> clos_trans y z -> clos_trans x z.(** Alternative definition by transitive extension on the left *)

  Inductive clos_trans_1n (x: A) : A -> Prop :=
    | t1n_step (y:A) : R x y -> clos_trans_1n x y
    | t1n_trans (y z:A) : R x y -> clos_trans_1n y z -> clos_trans_1n x z.(** Alternative definition by transitive extension on the left *)

  Inductive clos_trans_1n (x: A) : A -> Prop :=
    | t1n_step (y:A) : R x y -> clos_trans_1n x y
    | t1n_trans (y z:A) : R x y -> clos_trans_1n y z -> clos_trans_1n x z.(** Alternative definition by transitive extension on the left *)

  Inductive clos_trans_1n (x: A) : A -> Prop :=
    | t1n_step (y:A) : R x y -> clos_trans_1n x y
    | t1n_trans (y z:A) : R x y -> clos_trans_1n y z -> clos_trans_1n x z.(** Alternative definition by transitive extension on the right *)

  Inductive clos_trans_n1 (x: A) : A -> Prop :=
    | tn1_step (y:A) : R x y -> clos_trans_n1 x y
    | tn1_trans (y z:A) : R y z -> clos_trans_n1 x y -> clos_trans_n1 x z.(** Alternative definition by transitive extension on the right *)

  Inductive clos_trans_n1 (x: A) : A -> Prop :=
    | tn1_step (y:A) : R x y -> clos_trans_n1 x y
    | tn1_trans (y z:A) : R y z -> clos_trans_n1 x y -> clos_trans_n1 x z.(** Alternative definition by transitive extension on the right *)

  Inductive clos_trans_n1 (x: A) : A -> Prop :=
    | tn1_step (y:A) : R x y -> clos_trans_n1 x y
    | tn1_trans (y z:A) : R y z -> clos_trans_n1 x y -> clos_trans_n1 x z.(** Definition by direct transitive closure *)

  Inductive clos_refl (x: A) : A -> Prop :=
    | r_step (y:A) : R x y -> clos_refl x y
    | r_refl : clos_refl x x.(** Definition by direct transitive closure *)

  Inductive clos_refl (x: A) : A -> Prop :=
    | r_step (y:A) : R x y -> clos_refl x y
    | r_refl : clos_refl x x.(** Definition by direct transitive closure *)

  Inductive clos_refl (x: A) : A -> Prop :=
    | r_step (y:A) : R x y -> clos_refl x y
    | r_refl : clos_refl x x.(** Definition by direct reflexive-transitive closure *)

  Inductive clos_refl_trans (x:A) : A -> Prop :=
    | rt_step (y:A) : R x y -> clos_refl_trans x y
    | rt_refl : clos_refl_trans x x
    | rt_trans (y z:A) :
          clos_refl_trans x y -> clos_refl_trans y z -> clos_refl_trans x z.(** Definition by direct reflexive-transitive closure *)

  Inductive clos_refl_trans (x:A) : A -> Prop :=
    | rt_step (y:A) : R x y -> clos_refl_trans x y
    | rt_refl : clos_refl_trans x x
    | rt_trans (y z:A) :
          clos_refl_trans x y -> clos_refl_trans y z -> clos_refl_trans x z.(** Definition by direct reflexive-transitive closure *)

  Inductive clos_refl_trans (x:A) : A -> Prop :=
    | rt_step (y:A) : R x y -> clos_refl_trans x y
    | rt_refl : clos_refl_trans x x
    | rt_trans (y z:A) :
          clos_refl_trans x y -> clos_refl_trans y z -> clos_refl_trans x z.(** Definition by direct reflexive-transitive closure *)

  Inductive clos_refl_trans (x:A) : A -> Prop :=
    | rt_step (y:A) : R x y -> clos_refl_trans x y
    | rt_refl : clos_refl_trans x x
    | rt_trans (y z:A) :
          clos_refl_trans x y -> clos_refl_trans y z -> clos_refl_trans x z.(** Alternative definition by transitive extension on the left *)

  Inductive clos_refl_trans_1n (x: A) : A -> Prop :=
    | rt1n_refl : clos_refl_trans_1n x x
    | rt1n_trans (y z:A) :
         R x y -> clos_refl_trans_1n y z -> clos_refl_trans_1n x z.(** Alternative definition by transitive extension on the left *)

  Inductive clos_refl_trans_1n (x: A) : A -> Prop :=
    | rt1n_refl : clos_refl_trans_1n x x
    | rt1n_trans (y z:A) :
         R x y -> clos_refl_trans_1n y z -> clos_refl_trans_1n x z.(** Alternative definition by transitive extension on the left *)

  Inductive clos_refl_trans_1n (x: A) : A -> Prop :=
    | rt1n_refl : clos_refl_trans_1n x x
    | rt1n_trans (y z:A) :
         R x y -> clos_refl_trans_1n y z -> clos_refl_trans_1n x z.(** Alternative definition by transitive extension on the right *)

 Inductive clos_refl_trans_n1 (x: A) : A -> Prop :=
    | rtn1_refl : clos_refl_trans_n1 x x
    | rtn1_trans (y z:A) :
        R y z -> clos_refl_trans_n1 x y -> clos_refl_trans_n1 x z.(** Alternative definition by transitive extension on the right *)

 Inductive clos_refl_trans_n1 (x: A) : A -> Prop :=
    | rtn1_refl : clos_refl_trans_n1 x x
    | rtn1_trans (y z:A) :
        R y z -> clos_refl_trans_n1 x y -> clos_refl_trans_n1 x z.(** Alternative definition by transitive extension on the right *)

 Inductive clos_refl_trans_n1 (x: A) : A -> Prop :=
    | rtn1_refl : clos_refl_trans_n1 x x
    | rtn1_trans (y z:A) :
        R y z -> clos_refl_trans_n1 x y -> clos_refl_trans_n1 x z.(** Definition by direct reflexive-symmetric-transitive closure *)

  Inductive clos_refl_sym_trans : relation A :=
    | rst_step (x y:A) : R x y -> clos_refl_sym_trans x y
    | rst_refl (x:A) : clos_refl_sym_trans x x
    | rst_sym (x y:A) : clos_refl_sym_trans x y -> clos_refl_sym_trans y x
    | rst_trans (x y z:A) :
        clos_refl_sym_trans x y ->
        clos_refl_sym_trans y z -> clos_refl_sym_trans x z.(** Definition by direct reflexive-symmetric-transitive closure *)

  Inductive clos_refl_sym_trans : relation A :=
    | rst_step (x y:A) : R x y -> clos_refl_sym_trans x y
    | rst_refl (x:A) : clos_refl_sym_trans x x
    | rst_sym (x y:A) : clos_refl_sym_trans x y -> clos_refl_sym_trans y x
    | rst_trans (x y z:A) :
        clos_refl_sym_trans x y ->
        clos_refl_sym_trans y z -> clos_refl_sym_trans x z.(** Definition by direct reflexive-symmetric-transitive closure *)

  Inductive clos_refl_sym_trans : relation A :=
    | rst_step (x y:A) : R x y -> clos_refl_sym_trans x y
    | rst_refl (x:A) : clos_refl_sym_trans x x
    | rst_sym (x y:A) : clos_refl_sym_trans x y -> clos_refl_sym_trans y x
    | rst_trans (x y z:A) :
        clos_refl_sym_trans x y ->
        clos_refl_sym_trans y z -> clos_refl_sym_trans x z.(** Definition by direct reflexive-symmetric-transitive closure *)

  Inductive clos_refl_sym_trans : relation A :=
    | rst_step (x y:A) : R x y -> clos_refl_sym_trans x y
    | rst_refl (x:A) : clos_refl_sym_trans x x
    | rst_sym (x y:A) : clos_refl_sym_trans x y -> clos_refl_sym_trans y x
    | rst_trans (x y z:A) :
        clos_refl_sym_trans x y ->
        clos_refl_sym_trans y z -> clos_refl_sym_trans x z.(** Definition by direct reflexive-symmetric-transitive closure *)

  Inductive clos_refl_sym_trans : relation A :=
    | rst_step (x y:A) : R x y -> clos_refl_sym_trans x y
    | rst_refl (x:A) : clos_refl_sym_trans x x
    | rst_sym (x y:A) : clos_refl_sym_trans x y -> clos_refl_sym_trans y x
    | rst_trans (x y z:A) :
        clos_refl_sym_trans x y ->
        clos_refl_sym_trans y z -> clos_refl_sym_trans x z.(** Alternative definition by symmetric-transitive extension on the left *)

  Inductive clos_refl_sym_trans_1n (x: A) : A -> Prop :=
    | rst1n_refl : clos_refl_sym_trans_1n x x
    | rst1n_trans (y z:A) : R x y \/ R y x ->
         clos_refl_sym_trans_1n y z -> clos_refl_sym_trans_1n x z.(** Alternative definition by symmetric-transitive extension on the left *)

  Inductive clos_refl_sym_trans_1n (x: A) : A -> Prop :=
    | rst1n_refl : clos_refl_sym_trans_1n x x
    | rst1n_trans (y z:A) : R x y \/ R y x ->
         clos_refl_sym_trans_1n y z -> clos_refl_sym_trans_1n x z.(** Alternative definition by symmetric-transitive extension on the left *)

  Inductive clos_refl_sym_trans_1n (x: A) : A -> Prop :=
    | rst1n_refl : clos_refl_sym_trans_1n x x
    | rst1n_trans (y z:A) : R x y \/ R y x ->
         clos_refl_sym_trans_1n y z -> clos_refl_sym_trans_1n x z.(** Alternative definition by symmetric-transitive extension on the right *)

  Inductive clos_refl_sym_trans_n1 (x: A) : A -> Prop :=
    | rstn1_refl : clos_refl_sym_trans_n1 x x
    | rstn1_trans (y z:A) : R y z \/ R z y ->
         clos_refl_sym_trans_n1 x y -> clos_refl_sym_trans_n1 x z.(** Alternative definition by symmetric-transitive extension on the right *)

  Inductive clos_refl_sym_trans_n1 (x: A) : A -> Prop :=
    | rstn1_refl : clos_refl_sym_trans_n1 x x
    | rstn1_trans (y z:A) : R y z \/ R z y ->
         clos_refl_sym_trans_n1 x y -> clos_refl_sym_trans_n1 x z.(** Alternative definition by symmetric-transitive extension on the right *)

  Inductive clos_refl_sym_trans_n1 (x: A) : A -> Prop :=
    | rstn1_refl : clos_refl_sym_trans_n1 x x
    | rstn1_trans (y z:A) : R y z \/ R z y ->
         clos_refl_sym_trans_n1 x y -> clos_refl_sym_trans_n1 x z.Definition transp (x y:A) := R y x.Definition union (x y:A) := R1 x y \/ R2 x y.Inductive le_AsB : A + B -> A + B -> Prop :=
  | le_aa (x y:A) : leA x y -> le_AsB (inl _ x) (inl _ y)
  | le_ab (x:A) (y:B) : le_AsB (inl _ x) (inr _ y)
  | le_bb (x y:B) : leB x y -> le_AsB (inr _ x) (inr _ y).Inductive le_AsB : A + B -> A + B -> Prop :=
  | le_aa (x y:A) : leA x y -> le_AsB (inl _ x) (inl _ y)
  | le_ab (x:A) (y:B) : le_AsB (inl _ x) (inr _ y)
  | le_bb (x y:B) : leB x y -> le_AsB (inr _ x) (inr _ y).Inductive le_AsB : A + B -> A + B -> Prop :=
  | le_aa (x y:A) : leA x y -> le_AsB (inl _ x) (inl _ y)
  | le_ab (x:A) (y:B) : le_AsB (inl _ x) (inr _ y)
  | le_bb (x y:B) : leB x y -> le_AsB (inr _ x) (inr _ y).Inductive le_AsB : A + B -> A + B -> Prop :=
  | le_aa (x y:A) : leA x y -> le_AsB (inl _ x) (inl _ y)
  | le_ab (x:A) (y:B) : le_AsB (inl _ x) (inr _ y)
  | le_bb (x y:B) : leB x y -> le_AsB (inr _ x) (inr _ y).Inductive lexprod : sigT B -> sigT B -> Prop :=
    | left_lex :
      forall (x x' : A) (y : B x) (y' : B x'),
        leA x x' -> lexprod (x; y) (x'; y')
    | right_lex :
      forall (x : A) (y y' : B x),
        leB x y y' -> lexprod (x; y) (x; y').Inductive lexprod : sigT B -> sigT B -> Prop :=
    | left_lex :
      forall (x x' : A) (y : B x) (y' : B x'),
        leA x x' -> lexprod (x; y) (x'; y')
    | right_lex :
      forall (x : A) (y y' : B x),
        leB x y y' -> lexprod (x; y) (x; y').Inductive lexprod : sigT B -> sigT B -> Prop :=
    | left_lex :
      forall (x x' : A) (y : B x) (y' : B x'),
        leA x x' -> lexprod (x; y) (x'; y')
    | right_lex :
      forall (x : A) (y y' : B x),
        leB x y y' -> lexprod (x; y) (x; y').Inductive slexprod : A * B -> A * B -> Prop :=
    | left_slex :
      forall (x x' : A) (y : B) (y' : B),
        leA x x' -> slexprod (x, y) (x', y')
    | right_slex :
      forall (x : A) (y y' : B),
        leB y y' -> slexprod (x, y) (x, y').Inductive slexprod : A * B -> A * B -> Prop :=
    | left_slex :
      forall (x x' : A) (y : B) (y' : B),
        leA x x' -> slexprod (x, y) (x', y')
    | right_slex :
      forall (x : A) (y y' : B),
        leB y y' -> slexprod (x, y) (x, y').Inductive slexprod : A * B -> A * B -> Prop :=
    | left_slex :
      forall (x x' : A) (y : B) (y' : B),
        leA x x' -> slexprod (x, y) (x', y')
    | right_slex :
      forall (x : A) (y y' : B),
        leB y y' -> slexprod (x, y) (x, y').

(* Specifications: *)
Lemma slexprod_lexprod p1 p2 : slexprod p1 p2 <-> lexprod _ _ leA (fun _ => leB) (sigT_of_prod p1) (sigT_of_prod p2).