helm/matita/library/algebra/semigroups.ma

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  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    semigroup_properties: isSemiGroup carrier op
  26  }.
  27
  28 coercion cic:/matita/algebra/semigroups/carrier.con.
  29
  30 notation "hvbox(a break \middot \sub S b)"
  31   left associative with precedence 55
  32 for @{ 'ptimes $S $a $b }.
  33
  34 notation "hvbox(a break \middot b)"
  35   left associative with precedence 55
  36 for @{ 'ptimesi $a $b }.
  37
  38 interpretation "Semigroup operation" 'ptimesi a b =
  39  (cic:/matita/algebra/semigroups/op.con _ a b).
  40
  41 (* too ugly
  42 interpretation "Semigroup operation" 'ptimes S a b =
  43  (cic:/matita/algebra/semigroups/op.con S a b). *)
  44
  45 definition is_left_unit ≝
  46  λS:SemiGroup. λe:S. ∀x:S. e·x = x.
  47
  48 definition is_right_unit ≝
  49  λS:SemiGroup. λe:S. ∀x:S. x·e = x.
  50
  51 theorem is_left_unit_to_is_right_unit_to_eq:
  52  ∀S:SemiGroup. ∀e,e':S.
  53   is_left_unit ? e → is_right_unit ? e' → e=e'.
  54  intros;
  55  rewrite < (H e');
  56  rewrite < (H1 e) in \vdash (? ? % ?);
  57  reflexivity.
  58 qed.