From c48caa0775b4e9b95b27e20fca4bccb4e08134c2 Mon Sep 17 00:00:00 2001
From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Wed, 8 Jul 2009 15:53:26 +0000
Subject: [PATCH] ...

---
 .../matita/nlibrary/algebra/magmas.ma         | 54 +++++++++++++------
 1 file changed, 37 insertions(+), 17 deletions(-)

diff --git a/helm/software/matita/nlibrary/algebra/magmas.ma b/helm/software/matita/nlibrary/algebra/magmas.ma
index f58270a38..742a5ddb0 100644
--- a/helm/software/matita/nlibrary/algebra/magmas.ma
+++ b/helm/software/matita/nlibrary/algebra/magmas.ma
@@ -14,25 +14,46 @@
 
 include "sets/sets.ma".
 
-nrecord magma (A: Type) : Type[1] ≝
- { mcarr: Ω \sup A;
-   op: A → A → A;
-   op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op x y ∈ mcarr
+nrecord pre_magma : Type[1] ≝
+ { carr: Type;
+   op: carr → carr → carr
  }.
 (* this is a projection *)
-ndefinition mcarr ≝ λA.λM: magma A. match M with [ mk_magma mcarr _ _ ⇒ mcarr ].
-ndefinition op ≝ λA.λM: magma A. match M with [ mk_magma _ op _ ⇒ op ].
+ndefinition carr ≝ λM: pre_magma. match M with [ mk_pre_magma carr _ ⇒ carr ].
+ndefinition op ≝
+ λM: pre_magma. match M return λM. carr M → carr M → carr M with [ mk_pre_magma _ op ⇒ op ].
+(* ncoercion carr. *)
 
-(* to be splitted *)
-nrecord magma_morphism (A,B: Type) (Ma: magma A) (Mb: magma B) : Type ≝
- { mmcarr: A → B;
-   mmclosed: ∀x. x ∈ mcarr ? Ma → mmcarr x ∈ mcarr ? Mb;
-   (* need a canonical structure in next line? *)
-   mmprop: ∀x,y:A. x ∈ mcarr ? Ma → y ∈ mcarr ? Ma → mmcarr (op ? Ma x y) = op B Mb (mmcarr x) (mmcarr y)
+nrecord magma (A: pre_magma) : Type[1] ≝
+ { mcarr: Ω \sup (carr A);
+   op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
+ }.
+(* this is a projection *)
+ndefinition mcarr ≝ λA.λM: magma A. match M with [ mk_magma mcarr _ ⇒ mcarr ].
+ndefinition op_closed ≝
+ λA.λM: magma A.
+  match M return λM.∀x,y. x ∈ mcarr ? M → y ∈ mcarr ? M → op A x y ∈ mcarr ? M with
+   [ mk_magma _ opc ⇒ opc ].
+
+nrecord pre_magma_morphism (A,B: pre_magma) : Type ≝
+ { mmcarr: carr A → carr B;
+   mmprop: ∀x,y. mmcarr (op ? x y) = op ? (mmcarr x) (mmcarr y)
  }.
 (* this is a projection *)
 ndefinition mmcarr ≝
- λA,B,Ma,Mb.λf: magma_morphism A B Ma Mb. match f with [ mk_magma_morphism f _ _ ⇒ f ].
+ λA,B.λf: pre_magma_morphism A B. match f with [ mk_pre_magma_morphism f _ ⇒ f ].
+
+nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type ≝
+ { mmmcarr: pre_magma_morphism A B;
+   mmclosed: ∀x. x ∈ mcarr ? Ma → mmcarr ?? mmmcarr x ∈ mcarr ? Mb
+ }.
+(* this is a projection *)
+ndefinition mmmcarr ≝
+ λA,B,Ma,Mb.λf: magma_morphism A B Ma Mb. match f with [ mk_magma_morphism f _ ⇒ f ].
+ndefinition mmclosed ≝
+ λA,B,Ma,Mb.λf: magma_morphism A B Ma Mb.
+  match f return λf.∀x. x ∈ mcarr ? Ma → mmcarr ?? (mmmcarr ???? f) x ∈ mcarr ? Mb with
+   [ mk_magma_morphism _ p ⇒ p ].
  
 ndefinition sub_magma ≝
  λA.λM1,M2: magma A. ∀x. x ∈ mcarr ? M1 → x ∈ mcarr ? M2.
@@ -45,11 +66,10 @@ naxiom daemon: False.
 ndefinition mm_image:
  ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism ?? Ma Mb → magma B.
  #A; #B; #Ma; #Mb; #f;
- napply (mk_magma ????)
-  [ napply (image ?? (mmcarr ???? f) (mcarr ? Ma)) 
-  | napply (op ? Mb)
+ napply (mk_magma ???)
+  [ napply (image ?? (mmcarr ?? (mmmcarr ???? f)) (mcarr ? Ma)) 
   | #x; #y; *; #x0; #Hx0; *; #y0; #Hy0; nwhd;
     napply (ex_intro ????)
-     [ napply (op ? Ma x0 y0) (* BAD HERE! need a canonical structure? *) 
+     [ napply (op ? x0 y0) 
      | nelim daemon ]]
 nqed.
\ No newline at end of file
-- 
2.39.2