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

diff --git a/matita/library/algebra/semigroups.ma b/matita/library/algebra/semigroups.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/algebra/semigroups".
+
+include "higher_order_defs/functions.ma".
+
+(* Magmas *)
+
+record Magma : Type≝
+ { carrier:> Type;
+   op: carrier → carrier → carrier
+ }.
+
+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).
+
+(* Semigroups *)
+
+record isSemiGroup (M:Magma) : Prop≝
+ { op_associative: associative ? (op M) }.
+
+record SemiGroup : Type≝
+ { magma:> Magma;
+   semigroup_properties:> isSemiGroup magma
+ }.
+ 
+definition is_left_unit ≝
+ λS:SemiGroup. λe:S. ∀x:S. e·x = x.
+ 
+definition is_right_unit ≝
+ λS:SemiGroup. λe:S. ∀x:S. x·e = x.
+
+theorem is_left_unit_to_is_right_unit_to_eq:
+ ∀S:SemiGroup. ∀e,e':S.
+  is_left_unit ? e → is_right_unit ? e' → e=e'.
+ intros;
+ rewrite < (H e');
+ rewrite < (H1 e) in \vdash (? ? % ?).
+ reflexivity.
+qed.