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

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   3 (*      ||M||                                                             *)
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   5 (*      ||T||                                                             *)
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  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/class_eq".
  16
  17 include "class_defs.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; clear c2;
  21    [ auto | decompose H2; auto | decompose H2; auto ].
  22 qed.
  23
  24 theorem ceq_trans: \forall C,c2,c1,c3.
  25                    ceq C c2 c3 \to ceq ? c1 c2 \to ceq ? c1 c3.
  26 intros 5; elim H; clear H; clear c3;
  27    [ auto
  28    | apply ceq_sing_r; [||| apply H4 ]; auto
  29    | apply ceq_sing_l; [||| apply H4 ]; auto
  30    ].
  31 qed.
  32
  33 theorem ceq_conf_rev: \forall C,c2,c1,c3.
  34                       ceq C c3 c2 \to ceq ? c1 c2 \to ceq ? c1 c3.
  35 intros 5; elim H; clear H; clear c2;
  36    [ auto
  37    | lapply ceq_cl; [ decompose Hletin |||| apply H1 ].
  38      apply H2; apply ceq_sing_l; [||| apply H4 ]; auto
  39    | lapply ceq_cl; [ decompose Hletin |||| apply H1 ].
  40      apply H2; apply ceq_sing_r; [||| apply H4 ]; auto
  41    ].
  42 qed.
  43
  44 theorem ceq_sym: \forall C,c1,c2. ceq C c1 c2 \to ceq C c2 c1.
  45 intros;
  46 lapply ceq_cl; [ decompose Hletin |||| apply H ].
  47 auto.
  48 qed.
  49
  50 theorem ceq_conf: \forall C,c2,c1,c3.
  51                   ceq C c1 c2 \to ceq ? c1 c3 \to ceq ? c2 c3.
  52 intros.
  53 lapply ceq_sym; [|||| apply H ].
  54 apply ceq_trans; [| auto | auto ].
  55 qed.