Aggiornamento alla negazione.

[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 3424944bedea48628147f3165766185e2cd531b8..685d6248059afb3e162ae4ec00ea1304f05318af 100644 (file)
--- a/helm/software/matita/nlibrary/algebra/magmas.ma
+++ b/helm/software/matita/nlibrary/algebra/magmas.ma
@@ -14,8 +14,72 @@
 
 include "sets/sets.ma".
 
-nrecord magma (A: Type) : Type[1] ≝
- { carr: Ω \sup A;
-   op: A → A → A;
-   op_closed: ∀x,y. x ∈ carr → y ∈ carr → op x y ∈ carr
- }.
\ No newline at end of file
+nrecord magma_type : Type[1] ≝
+ { mtcarr:> setoid;
+   op: unary_morphism mtcarr (unary_morph_setoid mtcarr mtcarr)
+ }.
+
+nrecord magma (A: magma_type) : Type[1] ≝
+ { mcarr:> ext_powerclass A;
+   op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
+ }.
+
+alias symbol "hint_decl" = "hint_decl_Type2".
+unification hint 0 ≔ 
+  A : ? ⊢ carr1 (ext_powerclass_setoid A) ≡ ext_powerclass A. 
+
+(*
+ncoercion mcarr' : ∀A. ∀M: magma A. carr1 (qpowerclass_setoid (mtcarr A))
+ ≝ λA.λM: magma A.mcarr ? M
+ on _M: magma ? to carr1 (qpowerclass_setoid (mtcarr ?)).
+*)
+
+nrecord magma_morphism_type (A,B: magma_type) : Type[0] ≝
+ { mmcarr:> unary_morphism A B;
+   mmprop: ∀x,y:A. mmcarr (op ? x y) = op … (mmcarr x) (mmcarr y)
+ }.
+
+nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type[0] ≝
+ { mmmcarr:> magma_morphism_type A B;
+   mmclosed: ∀x:A. x ∈ mcarr ? Ma → mmmcarr x ∈ mcarr ? Mb
+ }.
+
+(*
+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 … 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
+        | nrewrite < Hx1;
+          nrewrite < Hy1;
+          napply (mmprop … f)]##]
+nqed.
+
+ndefinition mm_counter_image:
+ ∀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)
+   | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
+     napply ex_intro
+      [ napply (op … x0 y0)
+      | napply conj
+         [ napply op_closed; nassumption
+         | nrewrite < Hx1;
+           nrewrite < Hy1;
+           napply (mmprop … f)]##]
+nqed.
+*)
+
+ndefinition m_intersect: ∀A. magma A → magma A → magma A.
+ #A; #M1; #M2;
+ napply (mk_magma …)
+  [ napply (intersect_is_ext_morph ? M1 M2)
+  | #x; #y; nwhd in ⊢ (% → % → %); *; #Hx1; #Hx2; *; #Hy1; #Hy2;
+    napply conj; napply op_closed; nassumption ]
+nqed.
\ No newline at end of file