helm/matita/contribs/PREDICATIVE-TOPOLOGY/class_eq.ma

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/class_eq".
  16
  17 include "class_le.ma".
  18
  19 theorem ceq_cl: \forall C,c1,c2. ceq ? c1 c2 \to cin C c1 \land cin C c2.
  20 intros; elim H; clear H.
  21 lapply cle_cl to H1 using H; clear H1; decompose H;
  22 lapply cle_cl to H2 using H; clear H2; decompose H.
  23 auto.
  24 qed.
  25
  26 theorem ceq_refl: \forall C,c. cin C c \to ceq ? c c.
  27 intros; apply ceq_intro; auto.
  28 qed.
  29
  30 theorem ceq_trans: \forall C,c2,c1,c3.
  31                    ceq C c2 c3 \to ceq ? c1 c2 \to ceq ? c1 c3.
  32 intros; elim H; elim H1; clear H; clear H1.
  33 apply ceq_intro; apply cle_trans; [|auto|auto||auto|auto].
  34 qed.
  35
  36 theorem ceq_sym: \forall C,c1,c2. ceq C c1 c2 \to ceq C c2 c1.
  37 intros; elim H; clear H.; auto.
  38 qed.