matitaB/matita/library/algebra/semigroups.ma

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  14
  15 include "higher_order_defs/functions.ma".
  16
  17 (* Semigroups *)
  18
  19 record PreSemiGroup : Type≝
  20  { carrier:> Type;
  21    op: carrier → carrier → carrier
  22  }.
  23
  24 interpretation "Semigroup operation" 'middot a b = (op ? a b).
  25
  26 record IsSemiGroup (S:PreSemiGroup) : Prop ≝
  27  { op_is_associative: associative ? (op S) }.
  28
  29 record SemiGroup : Type≝
  30  { pre_semi_group:> PreSemiGroup;
  31    is_semi_group :> IsSemiGroup pre_semi_group
  32  }.
  33
  34 definition is_left_unit ≝
  35  λS:PreSemiGroup. λe:S. ∀x:S. e·x = x.
  36
  37 definition is_right_unit ≝
  38  λS:PreSemiGroup. λe:S. ∀x:S. x·e = x.
  39
  40 theorem is_left_unit_to_is_right_unit_to_eq:
  41  ∀S:PreSemiGroup. ∀e,e':S.
  42   is_left_unit ? e → is_right_unit ? e' → e=e'.
  43  intros;
  44  rewrite < (H e');
  45  rewrite < (H1 e) in \vdash (? ? % ?).
  46  reflexivity.
  47 qed.