...

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

diff --git a/helm/software/matita/library/algebra/semigroups.ma b/helm/software/matita/library/algebra/semigroups.ma
index 5b461d1a4f43edba6acc9e4dd448f0eceb157243..18d73b619402ecee81343ec6e9a9420181efa96f 100644 (file)
--- a/helm/software/matita/library/algebra/semigroups.ma
+++ b/helm/software/matita/library/algebra/semigroups.ma
@@ -12,53 +12,36 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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 ≝
- { 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.