we removed some coercion detours and we added some coercions we really would like...

[helm.git] / helm / software / matita / library / algebra / monoids.ma

diff --git a/helm/software/matita/library/algebra/monoids.ma b/helm/software/matita/library/algebra/monoids.ma
index 36de5a369d47eae97a210e6d0f7c8c64362e38a1..71e9a7c594a609d335a13501695cc657564881b9 100644 (file)
--- a/helm/software/matita/library/algebra/monoids.ma
+++ b/helm/software/matita/library/algebra/monoids.ma
@@ -15,33 +15,35 @@
 include "algebra/semigroups.ma".
 
 record PreMonoid : Type ≝
- { magma:> Magma;
-   e: magma
- }.
+{ pre_semi_group :> PreSemiGroup;
+  e              :  pre_semi_group 
+}.
 
-(* FG: the interpretation goes just after its definition *)
 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 
  }.
-  
+
+definition SemiGroup_of_Monoid: Monoid → SemiGroup ≝
+ λM. mk_SemiGroup ? (is_monoid M).
+
+coercion SemiGroup_of_Monoid nocomposites.
+
 definition is_left_inverse ≝
- λM:Monoid.
+ λM:PreMonoid.
   λopp: M → M.
    ∀x:M. (opp x)·x = ⅇ.
-   
+
 definition is_right_inverse ≝
- λM:Monoid.
+ λM:PreMonoid.
   λopp: M → M.
    ∀x:M. x·(opp x) = ⅇ.
 
@@ -50,15 +52,11 @@ theorem is_left_inverse_to_is_right_inverse_to_eq:
   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.