Setoids, setoids1, sets, and the like. The mess begins.

[helm.git] / helm / software / matita / nlibrary / algebra / magmas.ma

diff --git a/helm/software/matita/nlibrary/algebra/magmas.ma b/helm/software/matita/nlibrary/algebra/magmas.ma
index 1c829997aaf14724a02f5a39c2e22e411d06edf5..a879679a9feb760a2b7b30a979aeb3a6ed62f237 100644 (file)
--- a/helm/software/matita/nlibrary/algebra/magmas.ma
+++ b/helm/software/matita/nlibrary/algebra/magmas.ma
@@ -14,61 +14,66 @@
 
 include "sets/sets.ma".
 
-nrecord pre_magma : Type[1] ≝
- { carr:> Type;
-   op: carr → carr → carr
+nrecord magma_type : Type[1] ≝
+ { mcarr:> setoid;
+   op: mcarr → mcarr → mcarr
  }.
 
-nrecord magma (A: pre_magma) : Type[1] ≝
- { mcarr:> Ω \sup A;
+nrecord magma (A: magma_type) : Type[1] ≝
+ { mcarr:> powerset_setoid1 A;
    op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
  }.
 
-nrecord pre_magma_morphism (A,B: pre_magma) : Type ≝
+nrecord magma_morphism_type (A,B: magma_type) : Type ≝
  { mmcarr:1> A → B;
-   mmprop: ∀x,y. mmcarr (op ? x y) = op ? (mmcarr x) (mmcarr y)
+   mmprop: ∀x,y. mmcarr (op … x y) = op … (mmcarr x) (mmcarr y)
  }.
 
 nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type ≝
- { mmmcarr:> pre_magma_morphism A B;
+ { mmmcarr:> magma_morphism_type A B;
    mmclosed: ∀x. x ∈ Ma → mmmcarr x ∈ Mb
  }.
  
-ndefinition sub_magma ≝
- λA.λM1,M2: magma A. M1 ⊆ M2.
- 
 ndefinition image: ∀A,B. (A → B) → Ω \sup A → Ω \sup B ≝
  λA,B,f,Sa. {y | ∃x. x ∈ Sa ∧ f x = y}.
 
 ndefinition mm_image:
- ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism ?? Ma Mb → magma B.
+ ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma B.
  #A; #B; #Ma; #Mb; #f;
- napply (mk_magma ???)
-  [ napply (image ?? f Ma)
+ napply mk_magma
+  [ napply (image … f Ma)
   | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
-    napply (ex_intro ????)
-     [ napply (op ? x0 y0) 
-     | napply (conj ????)
-        [ napply (op_closed ??????); nassumption
+    napply ex_intro
+     [ napply (op … x0 y0) 
+     | napply conj
+        [ napply op_closed; nassumption
         | nrewrite < Hx1;
           nrewrite < Hy1;
-          napply (mmprop ?? f ??)]##]
+          napply (mmprop … f)]##]
 nqed.
 
 ndefinition counter_image: ∀A,B. (A → B) → Ω \sup B → Ω \sup A ≝
  λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
  
 ndefinition mm_counter_image:
- ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism ?? Ma Mb → magma A.
+ ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma A.
   #A; #B; #Ma; #Mb; #f;
-  napply (mk_magma ???)
-   [ napply (counter_image ?? f Mb)
+  napply mk_magma
+   [ napply (counter_image … f Mb)
    | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
-     napply (ex_intro ????)
-      [ napply (op ? x0 y0)
-      | napply (conj ????)
-         [ napply (op_closed ??????); nassumption
+     napply ex_intro
+      [ napply (op … x0 y0)
+      | napply conj
+         [ napply op_closed; nassumption
          | nrewrite < Hx1;
            nrewrite < Hy1;
-           napply (mmprop ?? f ??)]##]
+           napply (mmprop … f)]##]
+nqed.
+
+ndefinition m_intersect: ∀A. magma A → magma A → magma A.
+ #A; #M1; #M2;
+ napply (mk_magma …)
+  [ napply (M1 ∩ M2)
+  | #x; #y; nwhd in ⊢ (% → % → %); *; #Hx1; #Hx2; *; #Hy1; #Hy2;
+    napply conj; napply op_closed; nassumption ]
 nqed.
\ No newline at end of file