(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/algebra/semigroups".
-
include "higher_order_defs/functions.ma".
-(* Magmas *)
+(* Semigroups *)
-record Magma : Type ≝
+record PreSemiGroup : Type≝
{ carrier:> Type;
op: carrier → carrier → carrier
}.
-notation < "M" for @{ 'carrier $M }.
-interpretation "carrier coercion" 'carrier S =
- (cic:/matita/algebra/semigroups/carrier.con S).
-
-notation "hvbox(a break \middot b)"
- left associative with precedence 55
-for @{ 'magma_op $a $b }.
-
-interpretation "magma operation" 'magma_op a b =
- (cic:/matita/algebra/semigroups/op.con _ a b).
+interpretation "Semigroup operation" 'middot a b = (op ? a b).
-(* Semigroups *)
-
-record isSemiGroup (M:Magma) : Prop ≝
- { op_associative: associative ? (op M) }.
+record IsSemiGroup (S:PreSemiGroup) : Prop ≝
+ { op_is_associative: associative ? (op S) }.
-record SemiGroup : Type ≝
- { magma:> Magma;
- semigroup_properties:> isSemiGroup magma
+record SemiGroup : Type≝
+ { pre_semi_group:> PreSemiGroup;
+ is_semi_group :> IsSemiGroup pre_semi_group
}.
-
-notation < "S" for @{ 'magma $S }.
-interpretation "magma coercion" 'magma S =
- (cic:/matita/algebra/semigroups/magma.con S).
-
+
definition is_left_unit ≝
- λS:SemiGroup. λe:S. ∀x:S. e·x = x.
-
+ λS:PreSemiGroup. λe:S. ∀x:S. e·x = x.
+
definition is_right_unit ≝
- λS:SemiGroup. λe:S. ∀x:S. x·e = x.
+ λS:PreSemiGroup. λe:S. ∀x:S. x·e = x.
theorem is_left_unit_to_is_right_unit_to_eq:
- ∀S:SemiGroup. ∀e,e':S.
+ ∀S:PreSemiGroup. ∀e,e':S.
is_left_unit ? e → is_right_unit ? e' → e=e'.
intros;
rewrite < (H e');
- rewrite < (H1 e) in \vdash (? ? % ?);
+ rewrite < (H1 e) in \vdash (? ? % ?).
reflexivity.
qed.