removed no longer used METAs

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

diff --git a/helm/matita/library/algebra/monoids.ma b/helm/matita/library/algebra/monoids.ma
index c85d1ebc1842df1e80af57928ca936766ef83f46..c3f3cc48e8841b8139a915399030355744755038 100644 (file)
--- a/helm/matita/library/algebra/monoids.ma
+++ b/helm/matita/library/algebra/monoids.ma
@@ -16,58 +16,70 @@ set "baseuri" "cic:/matita/algebra/monoids/".
 
 include "algebra/semigroups.ma".
 
-record isMonoid (SS:SemiGroup) (e:SS) : Prop ≝
- { e_is_left_unit: is_left_unit SS e;
-   e_is_right_unit: is_right_unit SS e
+record PreMonoid : Type ≝
+ { magma:> Magma;
+   e: magma
+ }.
+
+notation < "M" for @{ 'pmmagma $M }.
+interpretation "premonoid magma coercion" 'pmmagma M =
+ (cic:/matita/algebra/monoids/magma.con M).
+
+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 Monoid : Type ≝
- { semigroup: SemiGroup;
-   e: semigroup;
-   properties: isMonoid ? e
+ { premonoid:> PreMonoid;
+   monoid_properties:> isMonoid premonoid 
  }.
- 
-coercion cic:/matita/algebra/monoids/semigroup.con.
-
-notation "hvbox(! \sub S)"
-for @{ 'munit $S }.
 
-interpretation "Monoid unit" 'munit S =
- (cic:/matita/algebra/monoids/e.con S).
+notation < "M" for @{ 'semigroup $M }.
+interpretation "premonoid coercion" 'premonoid M =
+ (cic:/matita/algebra/monoids/premonoid.con M).
  
-notation < "M"
-for @{ 'semigroup $M }.
-
-interpretation "Semigroup coercion" 'semigroup M =
- (cic:/matita/algebra/monoids/semigroup.con M).
+notation < "M" for @{ 'typeofmonoid $M }.
+interpretation "premonoid coercion" 'typeofmonoid M =
+ (cic:/matita/algebra/monoids/Type_of_Monoid.con M).
+ 
+notation < "M" for @{ 'magmaofmonoid $M }.
+interpretation "premonoid coercion" 'magmaofmonoid M =
+ (cic:/matita/algebra/monoids/Magma_of_Monoid.con M).
+ 
+notation "1" with precedence 89
+for @{ 'munit }.
 
+interpretation "Monoid unit" 'munit =
+ (cic:/matita/algebra/monoids/e.con _).
+  
 definition is_left_inverse ≝
  λM:Monoid.
   λopp: M → M.
-   ∀x:M. op M (opp x) x = ! \sub M.
-
+   ∀x:M. (opp x)·x = 1.
+   
 definition is_right_inverse ≝
  λM:Monoid.
   λopp: M → M.
-   ∀x:M. op M x (opp x) = ! \sub M.
+   ∀x:M. x·(opp x) = 1.
 
 theorem is_left_inverse_to_is_right_inverse_to_eq:
- ∀M:Monoid. ∀oppL,oppR.
-  is_left_inverse M oppL → is_right_inverse M oppR → 
-   ∀x:M. oppL x = oppR x.
+ ∀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. op M y (oppR x)) ? ? H2);
+ generalize in match (eq_f ? ? (λy.y·(r x)) ? ? H2);
  simplify; fold simplify (op M);
  intro; clear H2;
- generalize in match (properties (semigroup M));
- fold simplify (Type_of_Monoid M);
+ generalize in match (associative ? (is_semi_group ? (monoid_properties M)));
  intro;
- unfold isSemiGroup in H2; unfold associative in H2;
  rewrite > H2 in H3; clear H2;
  rewrite > H1 in H3;
- rewrite > (e_is_left_unit ? ? (properties M)) in H3;
- rewrite > (e_is_right_unit ? ? (properties M)) in H3;
+ rewrite > (e_is_left_unit ? (monoid_properties M)) in H3;
+ rewrite > (e_is_right_unit ? (monoid_properties M)) in H3;
  assumption.
 qed.
-