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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(****************************************************************************)
(*                                                                          *)
(*                         Naive Set Theory in Coq                          *)
(*                                                                          *)
(*                     INRIA                        INRIA                   *)
(*              Rocquencourt                        Sophia-Antipolis        *)
(*                                                                          *)
(*                                 Coq V6.1                                 *)
(*									    *)
(*			         Gilles Kahn 				    *)
(*				 Gerard Huet				    *)
(*									    *)
(*									    *)
(*                                                                          *)
(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks  *)
(* to the Newton Institute for providing an exceptional work environment    *)
(* in Summer 1995. Several developments by E. Ledinot were an inspiration.  *)
(****************************************************************************)

Require Export Ensembles.
Require Export Relations_1.

Section Partial_orders.
  Variable U : Type.

  Definition Carrier := Ensemble U.

  Definition Rel := Relation U.

  Record PO : Type := Definition_of_PO
    { Carrier_of : Ensemble U;
      Rel_of : Relation U;
      PO_cond1 : Inhabited U Carrier_of;
      PO_cond2 : Order U Rel_of }.
  Variable p : PO.

  Definition Strict_Rel_of : Rel := fun x y:U => Rel_of p x y /\ x <> y.

  Inductive covers (y x:U) : Prop :=
    Definition_of_covers :
    Strict_Rel_of x y ->
    ~ (exists z : _, Strict_Rel_of x z /\ Strict_Rel_of z y) ->
    covers y x.

End Partial_orders.

Hint Unfold Carrier_of Rel_of Strict_Rel_of: sets.
Hint Resolve Definition_of_covers: sets.


Section Partial_order_facts.
  Variable U : Type.
  Variable D : PO U.

  
U:Type
D:PO U
forall x y z : U,
Strict_Rel_of U D x y ->
Rel_of U D y z -> Strict_Rel_of U D x z

  
U:Type
D:PO U
forall x y z : U,
Strict_Rel_of U D x y ->
Rel_of U D y z -> Strict_Rel_of U D x z

    
U:Type
D:PO U
forall x y z : U,
Rel_of U D x y /\ x <> y ->
Rel_of U D y z -> Strict_Rel_of U D x z

    
U:Type
D:PO U
forall x y z : U,
Rel_of U D x y /\ x <> y ->
Rel_of U D y z -> Rel_of U D x z /\ x <> z

    
U:Type
D:PO U
forall (Carrier_of0 : Ensemble U)
  (Rel_of0 : Relation U),
Inhabited U Carrier_of0 ->
Order U Rel_of0 ->
forall x y z : U,
Rel_of0 x y /\ x <> y ->
Rel_of0 y z -> Rel_of0 x z /\ x <> z

    
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
Reflexive U R ->
Transitive U R ->
Antisymmetric U R ->
forall x y z : U,
R x y /\ x <> y -> R y z -> R x z /\ x <> z

    
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y /\ x <> y
H'5:R y z
R x z
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y /\ x <> y
H'5:R y z
x <> z

    
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y /\ x <> y
H'5:R y z
R x z
 apply H'2 with (y := y); tauto.
    
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y /\ x <> y
H'5:R y z
x <> z
 
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y /\ x <> y
H'5:R y z
H'6:x = z
False

      
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'5:R y z
H'6:x = z
H'7:R x y
H'8:x <> y
x = y

      
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'5:R y z
H'6:x = z
H'7:R x y
H'8:x <> y
R y x

      rewrite H'6; tauto.
  Qed.

  
U:Type
D:PO U
forall x y z : U,
Rel_of U D x y ->
Strict_Rel_of U D y z -> Strict_Rel_of U D x z

  
U:Type
D:PO U
forall x y z : U,
Rel_of U D x y ->
Strict_Rel_of U D y z -> Strict_Rel_of U D x z

    
U:Type
D:PO U
forall x y z : U,
Rel_of U D x y ->
Rel_of U D y z /\ y <> z -> Strict_Rel_of U D x z

    
U:Type
D:PO U
forall x y z : U,
Rel_of U D x y ->
Rel_of U D y z /\ y <> z -> Rel_of U D x z /\ x <> z

    
U:Type
D:PO U
forall (Carrier_of0 : Ensemble U)
  (Rel_of0 : Relation U),
Inhabited U Carrier_of0 ->
Order U Rel_of0 ->
forall x y z : U,
Rel_of0 x y ->
Rel_of0 y z /\ y <> z -> Rel_of0 x z /\ x <> z

    
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
Reflexive U R ->
Transitive U R ->
Antisymmetric U R ->
forall x y z : U,
R x y -> R y z /\ y <> z -> R x z /\ x <> z

    
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y
H'5:R y z /\ y <> z
R x z
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y
H'5:R y z /\ y <> z
x <> z

    
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y
H'5:R y z /\ y <> z
R x z
 apply H'2 with (y := y); tauto.
    
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y
H'5:R y z /\ y <> z
x <> z
 
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y
H'5:R y z /\ y <> z
H'6:x = z
False

      
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y
H'6:x = z
H'7:R y z
H'8:y <> z
y = z

      
U:Type
D:PO U
C:Ensemble U
R:Relation U
H':Inhabited U C
H'0:Order U R
H'1:Reflexive U R
H'2:Transitive U R
H'3:Antisymmetric U R
x, y, z:U
H'4:R x y
H'6:x = z
H'7:R y z
H'8:y <> z
R z y

      rewrite <- H'6; auto.
  Qed.

  
U:Type
D:PO U
Transitive U (Strict_Rel_of U D)

    
U:Type
D:PO U
forall x y z : U,
Strict_Rel_of U D x y ->
Strict_Rel_of U D y z -> Strict_Rel_of U D x z

    
U:Type
D:PO U
x, y, z:U
H':Strict_Rel_of U D x y
H'0:Strict_Rel_of U D y z
Strict_Rel_of U D x z

    apply Strict_Rel_Transitive_with_Rel with (y := y);
      [ intuition | unfold Strict_Rel_of in H', H'0; intuition ].
  Qed.
End Partial_order_facts.