helm/software/matita/library/algebra/semigroups.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "higher_order_defs/functions.ma".
  16
  17 (* Magmas *)
  18
  19 record Magma : Type≝
  20  { carrier:> Type;
  21    op: carrier → carrier → carrier
  22  }.
  23
  24 interpretation "magma operation" 'middot a b = (op ? a b).
  25
  26 (* Semigroups *)
  27
  28 record isSemiGroup (M:Magma) : Prop≝
  29  { op_associative: associative ? (op M) }.
  30
  31 record SemiGroup : Type≝
  32  { magma:> Magma;
  33    semigroup_properties:> isSemiGroup magma
  34  }.
  35
  36 definition is_left_unit ≝
  37  λS:SemiGroup. λe:S. ∀x:S. e·x = x.
  38
  39 definition is_right_unit ≝
  40  λS:SemiGroup. λe:S. ∀x:S. x·e = x.
  41
  42 theorem is_left_unit_to_is_right_unit_to_eq:
  43  ∀S:SemiGroup. ∀e,e':S.
  44   is_left_unit ? e → is_right_unit ? e' → e=e'.
  45  intros;
  46  rewrite < (H e');
  47  rewrite < (H1 e) in \vdash (? ? % ?).
  48  reflexivity.
  49 qed.