- renamed ocaml/ to components/

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

diff --git a/helm/matita/library/algebra/monoids.ma b/helm/matita/library/algebra/monoids.ma
deleted file mode 100644 (file)
index c3f3cc4..0000000
--- a/helm/matita/library/algebra/monoids.ma
+++ /dev/null
@@ -1,85 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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 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 ≝
- { premonoid:> PreMonoid;
-   monoid_properties:> isMonoid premonoid 
- }.
-
-notation < "M" for @{ 'semigroup $M }.
-interpretation "premonoid coercion" 'premonoid M =
- (cic:/matita/algebra/monoids/premonoid.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. (opp x)·x = 1.
-   
-definition is_right_inverse ≝
- λM:Monoid.
-  λopp: M → M.
-   ∀x:M. x·(opp x) = 1.
-
-theorem is_left_inverse_to_is_right_inverse_to_eq:
- ∀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.y·(r x)) ? ? H2);
- simplify; fold simplify (op M);
- intro; clear H2;
- generalize in match (associative ? (is_semi_group ? (monoid_properties M)));
- intro;
- rewrite > H2 in H3; clear H2;
- rewrite > H1 in H3;
- rewrite > (e_is_left_unit ? (monoid_properties M)) in H3;
- rewrite > (e_is_right_unit ? (monoid_properties M)) in H3;
- assumption.
-qed.