theories/Relations

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+
+(* $Id$ *)
+
+(****************************************************************************)
+(*                            Bruno Barras                                  *)
+(****************************************************************************)
+
+Require Relation_Definitions.
+Require Relation_Operators.
+
+
+Section Properties.
+
+  Variable A: Set.
+  Variable R: (relation A).
+
+  Local incl : (relation A)->(relation A)->Prop :=
+      [R1,R2: (relation A)] (x,y:A) (R1 x y) -> (R2 x y).
+
+Section Clos_Refl_Trans.
+
+  Lemma clos_rt_is_preorder: (preorder A (clos_refl_trans A R)).
+Apply Build_preorder.
+Exact (rt_refl A R).
+
+Exact (rt_trans A R).
+Save.
+
+
+
+Lemma clos_rt_idempotent:
+       (incl (clos_refl_trans A (clos_refl_trans A R))
+                     (clos_refl_trans A R)).
+Red.
+Induction 1; Auto with sets.
+Intros.
+Apply rt_trans with y0; Auto with sets.
+Save.
+
+  Lemma  clos_refl_trans_ind_left: (A:Set)(R:A->A->Prop)(M:A)(P:A->Prop)
+           (P M)
+            ->((P0,N:A)
+                (clos_refl_trans A R M P0)->(P P0)->(R P0 N)->(P N))
+               ->(a:A)(clos_refl_trans A R M a)->(P a).
+Intros.
+Generalize H H0 .
+Clear  H H0.
+(Elim H1; Intros; Auto with sets).
+(Apply H2 with x; Auto with sets).
+
+Apply H3.
+(Apply H0; Auto with sets).
+
+Intros.
+(Apply H5 with P0; Auto with sets).
+(Apply rt_trans with y; Auto with sets).
+Save.
+
+
+End Clos_Refl_Trans.
+
+
+Section Clos_Refl_Sym_Trans.
+
+  Lemma clos_rt_clos_rst: (inclusion A (clos_refl_trans A R)
+				        (clos_refl_sym_trans A R)).
+Red.
+(Induction 1; Auto with sets).
+Intros.
+(Apply rst_trans with y0; Auto with sets).
+Save.
+
+  Lemma clos_rst_is_equiv: (equivalence A (clos_refl_sym_trans A R)).
+Apply Build_equivalence.
+Exact (rst_refl A R).
+
+Exact (rst_trans A R).
+
+Exact (rst_sym A R).
+Save.
+
+  Lemma clos_rst_idempotent:
+       (incl (clos_refl_sym_trans A (clos_refl_sym_trans A R))
+                     (clos_refl_sym_trans A R)).
+Red.
+(Induction 1; Auto with sets).
+Intros.
+(Apply rst_trans with y0; Auto with sets).
+Save.
+
+End Clos_Refl_Sym_Trans.
+
+End Properties.