An embryonic kernel of the algebraic library.

[helm.git] / helm / matita / library / algebra / monoids.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/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 Monoid : Type ≝
+ { semigroup: SemiGroup;
+   e: semigroup;
+   properties: isMonoid ? e
+ }.
+ 
+coercion cic:/matita/algebra/monoids/semigroup.con.
+
+definition is_left_inverse ≝
+ λM:Monoid. let op ≝ op M in let e ≝ e M in
+  λopp: M → M.
+   ∀x:M. op (opp x) x = e.
+
+definition is_right_inverse ≝
+ λM:Monoid. let op ≝ op M in let e ≝ e M in
+  λopp: M → M.
+   ∀x:M. op x (opp x) = e.
+
+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.
+ intros;
+ letin op ≝ (op M);
+ letin e ≝ (e M);
+ generalize in match (H x); intro;
+ change in H2 with (op (oppL x) x = e); 
+ generalize in match (eq_f ? ? (λy. op y (oppR x)) ? ? H2);
+ simplify; fold simplify op;
+ intro; clear H2;
+ generalize in match (properties (semigroup 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;
+ assumption.
+qed.
+