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 f58270a3889d027538bb38bd33627292b71aa8de..a879679a9feb760a2b7b30a979aeb3a6ed62f237 100644 (file)
--- a/helm/software/matita/nlibrary/algebra/magmas.ma
+++ b/helm/software/matita/nlibrary/algebra/magmas.ma
@@ -14,42 +14,66 @@
 
 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 magma_type : Type[1] ≝
+ { mcarr:> setoid;
+   op: mcarr → mcarr → mcarr
  }.
-(* 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 ].
-
-(* 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: magma_type) : Type[1] ≝
+ { mcarr:> powerset_setoid1 A;
+   op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
+ }.
+
+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)
+ }.
+
+nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type ≝
+ { mmmcarr:> magma_morphism_type A B;
+   mmclosed: ∀x. x ∈ Ma → mmmcarr x ∈ Mb
  }.
-(* 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 ].
- 
-ndefinition sub_magma ≝
- λA.λM1,M2: magma A. ∀x. x ∈ mcarr ? M1 → x ∈ mcarr ? M2.
  
 ndefinition image: ∀A,B. (A → B) → Ω \sup A → Ω \sup B ≝
  λA,B,f,Sa. {y | ∃x. x ∈ Sa ∧ f x = y}.
 
-naxiom daemon: False.
-
 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 ?? (mmcarr ???? f) (mcarr ? Ma)) 
-  | napply (op ? Mb)
-  | #x; #y; *; #x0; #Hx0; *; #y0; #Hy0; nwhd;
-    napply (ex_intro ????)
-     [ napply (op ? Ma x0 y0) (* BAD HERE! need a canonical structure? *) 
-     | nelim daemon ]]
+ 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 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; #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 (M1 ∩ M2)
+  | #x; #y; nwhd in ⊢ (% → % → %); *; #Hx1; #Hx2; *; #Hy1; #Hy2;
+    napply conj; napply op_closed; nassumption ]
 nqed.
\ No newline at end of file