X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Falgebra%2Fgroups.ma;h=ae6b525e3906d9b6f5b153b0c767ed6c47faed08;hb=ec13ac23f555f04b0a546d0e8ae464d9b5806d9b;hp=15361b6c9a1a93ea6b498b5e57e324aedc03b04a;hpb=dbd36ab2cf360a7721874e32654831ca3aa2f1cf;p=helm.git

diff --git a/helm/matita/library/algebra/groups.ma b/helm/matita/library/algebra/groups.ma
index 15361b6c9..ae6b525e3 100644
--- a/helm/matita/library/algebra/groups.ma
+++ b/helm/matita/library/algebra/groups.ma
@@ -17,24 +17,27 @@ set "baseuri" "cic:/matita/algebra/groups/".
 include "algebra/monoids.ma".
 include "nat/le_arith.ma".
 
-record isGroup (M:Monoid) (opp: M -> M) : Prop ≝
- { opp_is_left_inverse: is_left_inverse M opp;
-   opp_is_right_inverse: is_right_inverse M opp
+record PreGroup : Type ≝
+ { premonoid:> PreMonoid;
+   opp: premonoid -> premonoid
+ }.
+
+record isGroup (G:PreGroup) : Prop ≝
+ { is_monoid: isMonoid G;
+   opp_is_left_inverse: is_left_inverse (mk_Monoid ? is_monoid) (opp G);
+   opp_is_right_inverse: is_right_inverse (mk_Monoid ? is_monoid) (opp G)
  }.
  
 record Group : Type ≝
- { monoid: Monoid;
-   opp: monoid -> monoid;
-   group_properties: isGroup ? opp
+ { pregroup:> PreGroup;
+   group_properties:> isGroup pregroup
  }.
 
-coercion cic:/matita/algebra/groups/monoid.con.
-
-notation < "G"
+(*notation < "G"
 for @{ 'monoid $G }.
 
 interpretation "Monoid coercion" 'monoid G =
- (cic:/matita/algebra/groups/monoid.con G).
+ (cic:/matita/algebra/groups/monoid.con G).*)
 
 notation < "G"
 for @{ 'type_of_group $G }.
@@ -43,10 +46,10 @@ interpretation "Type_of_group coercion" 'type_of_group G =
  (cic:/matita/algebra/groups/Type_of_Group.con G).
 
 notation < "G"
-for @{ 'semigroup_of_group $G }.
+for @{ 'magma_of_group $G }.
 
-interpretation "Semigroup_of_group coercion" 'semigroup_of_group G =
- (cic:/matita/algebra/groups/SemiGroup_of_Group.con G).
+interpretation "magma_of_group coercion" 'magma_of_group G =
+ (cic:/matita/algebra/groups/Magma_of_Group.con G).
 
 notation "hvbox(x \sup (-1))" with precedence 89
 for @{ 'gopp $x }.
@@ -68,11 +71,11 @@ intros;
 unfold left_cancellable;
 unfold injective;
 intros (x y z);
-rewrite < (e_is_left_unit ? ? (monoid_properties G));
-rewrite < (e_is_left_unit ? ? (monoid_properties G) z);
-rewrite < (opp_is_left_inverse ? ? (group_properties G) x);
-rewrite > (semigroup_properties G);
-rewrite > (semigroup_properties G);
+rewrite < (e_is_left_unit ? (is_monoid ? (group_properties G)));
+rewrite < (e_is_left_unit ? (is_monoid ? (group_properties G)) z);
+rewrite < (opp_is_left_inverse ? (group_properties G) x);
+rewrite > (associative ? (is_semi_group ? (is_monoid ? (group_properties G))));
+rewrite > (associative ? (is_semi_group ? (is_monoid ? (group_properties G))));
 apply eq_f;
 assumption.
 qed.
@@ -85,11 +88,11 @@ unfold right_cancellable;
 unfold injective;
 simplify;fold simplify (op G); 
 intros (x y z);
-rewrite < (e_is_right_unit ? ? (monoid_properties G));
-rewrite < (e_is_right_unit ? ? (monoid_properties G) z);
-rewrite < (opp_is_right_inverse ? ? (group_properties G) x);
-rewrite < (semigroup_properties G);
-rewrite < (semigroup_properties G);
+rewrite < (e_is_right_unit ? (is_monoid ? (group_properties G)));
+rewrite < (e_is_right_unit ? (is_monoid ? (group_properties G)) z);
+rewrite < (opp_is_right_inverse ? (group_properties G) x);
+rewrite < (associative ? (is_semi_group ? (is_monoid ? (group_properties G))));
+rewrite < (associative ? (is_semi_group ? (is_monoid ? (group_properties G))));
 rewrite > H;
 reflexivity.
 qed.
@@ -103,70 +106,106 @@ record finite_enumerable (T:Type) : Type ≝
    index_of_repr: ∀n. n≤order → index_of (repr n) = n;
    repr_index_of: ∀x. repr (index_of x) = x
  }.
-
-notation < "hvbox(C \sub i)" with precedence 89
+ 
+notation "hvbox(C \sub i)" with precedence 89
 for @{ 'repr $C $i }.
 
+(* CSC: multiple interpretations in the same file are not considered in the
+ right order
 interpretation "Finite_enumerable representation" 'repr C i =
- (cic:/matita/algebra/groups/repr.con C _ i).
+ (cic:/matita/algebra/groups/repr.con C _ i).*)
  
-notation < "hvbox(|C|)" with precedence 89
+notation "hvbox(|C|)" with precedence 89
 for @{ 'card $C }.
 
 interpretation "Finite_enumerable order" 'card C =
  (cic:/matita/algebra/groups/order.con C _).
 
-theorem repr_inj:
- ∀T:Type. ∀H:finite_enumerable T.
-  ∀n,n'. n ≤ order ? H → n' ≤ order ? H →
-   repr ? H n = repr ? H n' → n=n'.
-intros;
-rewrite < (index_of_repr ? ? ? H1);
-rewrite > H3;
-apply index_of_repr;
-assumption.
-qed.
+record finite_enumerable_SemiGroup : Type ≝
+ { semigroup:> SemiGroup;
+   is_finite_enumerable:> finite_enumerable semigroup
+ }.
+
+notation < "S"
+for @{ 'semigroup_of_finite_enumerable_semigroup $S }.
+
+interpretation "Semigroup_of_finite_enumerable_semigroup"
+ 'semigroup_of_finite_enumerable_semigroup S
+=
+ (cic:/matita/algebra/groups/semigroup.con S).
 
+notation < "S"
+for @{ 'magma_of_finite_enumerable_semigroup $S }.
+
+interpretation "Magma_of_finite_enumerable_semigroup"
+ 'magma_of_finite_enumerable_semigroup S
+=
+ (cic:/matita/algebra/groups/Magma_of_finite_enumerable_SemiGroup.con S).
+ 
+notation < "S"
+for @{ 'type_of_finite_enumerable_semigroup $S }.
+
+interpretation "Type_of_finite_enumerable_semigroup"
+ 'type_of_finite_enumerable_semigroup S
+=
+ (cic:/matita/algebra/groups/Type_of_finite_enumerable_SemiGroup.con S).
+
+interpretation "Finite_enumerable representation" 'repr S i =
+ (cic:/matita/algebra/groups/repr.con S
+  (cic:/matita/algebra/groups/is_finite_enumerable.con S) i).
+
+notation "hvbox(ι e)" with precedence 60
+for @{ 'index_of_finite_enumerable_semigroup $e }.
+
+interpretation "Index_of_finite_enumerable representation"
+ 'index_of_finite_enumerable_semigroup e
+=
+ (cic:/matita/algebra/groups/index_of.con _
+  (cic:/matita/algebra/groups/is_finite_enumerable.con _) e).
+ 
 theorem foo:
- ∀G:SemiGroup.
-  finite_enumerable (carrier G) →
-   left_cancellable (carrier G) (op G) →
-    right_cancellable (carrier G) (op G) →
-     ∃e:G. isMonoid ? e.
-intros (G H);
-letin f ≝ (λn.index_of ? H ((repr ? H O)·(repr ? H n)));
-cut (∀n.n ≤ order ? H → ∃m.f m = n);
+ ∀G:finite_enumerable_SemiGroup.
+  left_cancellable ? (op G) →
+  right_cancellable ? (op G) →
+   ∃e:G. isMonoid (mk_PreMonoid G e).
+intros;
+letin f ≝ (λn.ι(G \sub O · G \sub n));
+cut (∀n.n ≤ order ? (is_finite_enumerable G) → ∃m.f m = n);
 [ letin EX ≝ (Hcut O ?);
   [ apply le_O_n
   | clearbody EX;
     clear Hcut;
     unfold f in EX;
     elim EX;
-    letin HH ≝ (eq_f ? ? (repr ? H) ? ? H3);
+    clear EX;
+    letin HH ≝ (eq_f ? ? (repr ? (is_finite_enumerable G)) ? ? H2);
     clearbody HH;
-    rewrite > (repr_index_of ? H) in HH;
-    apply (ex_intro ? ? (repr ? H a));
-    letin GOGO ≝ (refl_eq ? (repr ? H O));
+    rewrite > (repr_index_of ? (is_finite_enumerable G)) in HH;
+    apply (ex_intro ? ? (G \sub a));
+    letin GOGO ≝ (refl_eq ? (repr ? (is_finite_enumerable G) O));
     clearbody GOGO;
     rewrite < HH in GOGO;
     rewrite < HH in GOGO:(? ? % ?);
-    rewrite > (semigroup_properties G) in GOGO;
-    letin GaGa ≝ (H1 ? ? ? GOGO);
+    rewrite > (associative ? G) in GOGO;
+    letin GaGa ≝ (H ? ? ? GOGO);
     clearbody GaGa;
     clear GOGO;
     constructor 1;
-    [ unfold is_left_unit; intro;
-      letin GaxGax ≝ (refl_eq ? ((repr ? H a)·x));
+    [ simplify;
+      apply (semigroup_properties G)
+    | unfold is_left_unit; intro;
+      letin GaxGax ≝ (refl_eq ? (G \sub a ·x));
       clearbody GaxGax;
       rewrite < GaGa in GaxGax:(? ? % ?);
-      rewrite > (semigroup_properties G) in GaxGax;
-      apply (H1 ? ? ? GaxGax)
+      rewrite > (associative ? (semigroup_properties G)) in GaxGax;
+      apply (H ? ? ? GaxGax)
     | unfold is_right_unit; intro;
-      letin GaxGax ≝ (refl_eq ? (x·(repr ? H a)));
+      letin GaxGax ≝ (refl_eq ? (x·G \sub a));
       clearbody GaxGax;
       rewrite < GaGa in GaxGax:(? ? % ?);
-      rewrite < (semigroup_properties G) in GaxGax;
-      apply (H2 ? ? ? GaxGax)
+      rewrite < (associative ? (semigroup_properties G)) in GaxGax;
+      apply (H1 ? ? ? GaxGax)
+    ]
   ]
 |
-].
\ No newline at end of file
+].