X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Falgebra%2Fmonoids.ma;h=71e9a7c594a609d335a13501695cc657564881b9;hb=bee436af0c6ceb1c83259c94036df8b12f901f2d;hp=556faf6caf20445fe37d18f9fb7b780f7df62f10;hpb=9eabe046c1182960de8cfdba96c5414224e3a61e;p=helm.git

diff --git a/helm/software/matita/library/algebra/monoids.ma b/helm/software/matita/library/algebra/monoids.ma
index 556faf6ca..71e9a7c59 100644
--- a/helm/software/matita/library/algebra/monoids.ma
+++ b/helm/software/matita/library/algebra/monoids.ma
@@ -15,53 +15,48 @@
 include "algebra/semigroups.ma".
 
 record PreMonoid : Type ≝
- { magma:> Magma;
-   e: magma
- }.
+{ pre_semi_group :> PreSemiGroup;
+  e              :  pre_semi_group 
+}.
+
+interpretation "Monoid unit" 'neutral = (e ?).
 
-record isMonoid (M:PreMonoid) : Prop ≝
- { is_semi_group:> isSemiGroup M;
-   e_is_left_unit:
-    is_left_unit (mk_SemiGroup ? is_semi_group) (e M);
-   e_is_right_unit:
-    is_right_unit (mk_SemiGroup ? is_semi_group) (e M)
+record IsMonoid (M:PreMonoid): Prop ≝
+ { is_pre_semi_group :> IsSemiGroup M;
+   e_is_left_unit    :  is_left_unit M ⅇ;
+   e_is_right_unit   :  is_right_unit M ⅇ
  }.
- 
+
 record Monoid : Type ≝
- { premonoid:> PreMonoid;
-   monoid_properties:> isMonoid premonoid 
+ { pre_monoid :> PreMonoid;
+   is_monoid  :> IsMonoid pre_monoid 
  }.
 
-notation "1" with precedence 89
-for @{ 'munit }.
+definition SemiGroup_of_Monoid: Monoid → SemiGroup ≝
+ λM. mk_SemiGroup ? (is_monoid M).
+
+coercion SemiGroup_of_Monoid nocomposites.
 
-interpretation "Monoid unit" 'munit =
- (cic:/matita/algebra/monoids/e.con _).
-  
 definition is_left_inverse ≝
- λM:Monoid.
+ λM:PreMonoid.
   λopp: M → M.
-   ∀x:M. (opp x)·x = 1.
-   
+   ∀x:M. (opp x)·x = ⅇ.
+
 definition is_right_inverse ≝
- λM:Monoid.
+ λM:PreMonoid.
   λopp: M → M.
-   ∀x:M. x·(opp x) = 1.
+   ∀x:M. x·(opp x) = ⅇ.
 
 theorem is_left_inverse_to_is_right_inverse_to_eq:
  ∀M:Monoid. ∀l,r.
   is_left_inverse M l → is_right_inverse M r → 
    ∀x:M. l x = r x.
  intros;
- generalize in match (H x); intro;
- generalize in match (eq_f ? ? (λy.y·(r x)) ? ? H2);
- simplify; fold simplify (op M);
- intro; clear H2;
- generalize in match (op_associative ? (is_semi_group ? (monoid_properties M)));
- intro;
- rewrite > H2 in H3; clear H2;
+ lapply (H x) as H2;
+ lapply (eq_f ? ? (λy.y·(r x)) ? ? H2) as H3; clear H2;
+ rewrite > (op_is_associative ? M) in H3.
  rewrite > H1 in H3;
- rewrite > (e_is_left_unit ? (monoid_properties M)) in H3;
- rewrite > (e_is_right_unit ? (monoid_properties M)) in H3;
+ rewrite > (e_is_left_unit ? M) in H3;
+ rewrite > (e_is_right_unit ? M) in H3;
  assumption.
 qed.