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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   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 < "M" for @{ 'carrier $M }.
  27 interpretation "carrier coercion" 'carrier S =
  28  (cic:/matita/algebra/semigroups/carrier.con S).
  29
  30 notation "hvbox(a break \middot b)"
  31   left associative with precedence 55
  32 for @{ 'magma_op $a $b }.
  33
  34 interpretation "magma operation" 'magma_op a b =
  35  (cic:/matita/algebra/semigroups/op.con _ a b).
  36
  37 (* Semigroups *)
  38
  39 record isSemiGroup (M:Magma) : Prop ≝
  40  { associative: associative ? (op M) }.
  41
  42 record SemiGroup : Type ≝
  43  { magma:> Magma;
  44    semigroup_properties:> isSemiGroup magma
  45  }.
  46
  47 notation < "S" for @{ 'magma $S }.
  48 interpretation "magma coercion" 'magma S =
  49  (cic:/matita/algebra/semigroups/magma.con S).
  50
  51 definition is_left_unit ≝
  52  λS:SemiGroup. λe:S. ∀x:S. e·x = x.
  53
  54 definition is_right_unit ≝
  55  λS:SemiGroup. λe:S. ∀x:S. x·e = x.
  56
  57 theorem is_left_unit_to_is_right_unit_to_eq:
  58  ∀S:SemiGroup. ∀e,e':S.
  59   is_left_unit ? e → is_right_unit ? e' → e=e'.
  60  intros;
  61  rewrite < (H e');
  62  rewrite < (H1 e) in \vdash (? ? % ?);
  63  reflexivity.
  64 qed.