helm/matita/library/algebra/semigroups.ma

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   3 (*      ||M||                                                             *)
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  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/algebra/semigroups".
  16
  17 include "higher_order_defs/functions.ma".
  18
  19 definition isSemiGroup ≝
  20  λC:Type. λop: C → C → C.associative C op.
  21
  22 record SemiGroup : Type ≝
  23  { carrier: Type;
  24    op: carrier → carrier → carrier;
  25    properties: isSemiGroup carrier op
  26  }.
  27
  28 coercion cic:/matita/algebra/semigroups/carrier.con.
  29
  30 definition is_left_unit ≝
  31  λS:SemiGroup. λe:S. ∀x:S. op S e x = x.
  32
  33 definition is_right_unit ≝
  34  λS:SemiGroup. λe:S. ∀x:S. op S x e = x.
  35
  36 theorem is_left_unit_to_is_right_unit_to_eq:
  37  ∀S:SemiGroup. ∀e1,e2:S.
  38   is_left_unit ? e1 → is_right_unit ? e2 → e1=e2.
  39  intros;
  40  rewrite < (H e2);
  41  rewrite < (H1 e1) in \vdash (? ? % ?);
  42  reflexivity.
  43 qed.