matita/library/algebra/semigroups.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/algebra/semigroups".
  16
  17 include "higher_order_defs/functions.ma".
  18
  19 (* Magmas *)
  20
  21 record Magma : Type≝
  22  { carrier:> Type;
  23    op: carrier → carrier → carrier
  24  }.
  25
  26 notation "hvbox(a break \middot b)"
  27   left associative with precedence 55
  28 for @{ 'magma_op $a $b }.
  29
  30 interpretation "magma operation" 'magma_op a b =
  31  (cic:/matita/algebra/semigroups/op.con _ a b).
  32
  33 (* Semigroups *)
  34
  35 record isSemiGroup (M:Magma) : Prop≝
  36  { op_associative: associative ? (op M) }.
  37
  38 record SemiGroup : Type≝
  39  { magma:> Magma;
  40    semigroup_properties:> isSemiGroup magma
  41  }.
  42
  43 definition is_left_unit ≝
  44  λS:SemiGroup. λe:S. ∀x:S. e·x = x.
  45
  46 definition is_right_unit ≝
  47  λS:SemiGroup. λe:S. ∀x:S. x·e = x.
  48
  49 theorem is_left_unit_to_is_right_unit_to_eq:
  50  ∀S:SemiGroup. ∀e,e':S.
  51   is_left_unit ? e → is_right_unit ? e' → e=e'.
  52  intros;
  53  rewrite < (H e');
  54  rewrite < (H1 e) in \vdash (? ? % ?).
  55  reflexivity.
  56 qed.