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 de8e3ba22449f445dc210bfb66392649b11a9276..a879679a9feb760a2b7b30a979aeb3a6ed62f237 100644 (file)
--- a/helm/software/matita/nlibrary/algebra/magmas.ma
+++ b/helm/software/matita/nlibrary/algebra/magmas.ma
@@ -14,8 +14,66 @@
 
 include "sets/sets.ma".
 
-nrecord magma (A: Type) : Type ≝
- { 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] ≝
+ { mcarr:> setoid;
+   op: mcarr → mcarr → mcarr
+ }.
+
+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
+ }.
+ 
+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; #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 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