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 40400378dc7e1bedf9b2480823f9127a93df7d2b..a879679a9feb760a2b7b30a979aeb3a6ed62f237 100644 (file)
--- a/helm/software/matita/nlibrary/algebra/magmas.ma
+++ b/helm/software/matita/nlibrary/algebra/magmas.ma
@@ -14,76 +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
  }.
-(* this is a projection *)
-ndefinition carr: pre_magma → Type
- ≝ λM: pre_magma. match M with [ mk_pre_magma carr _ ⇒ carr ].
-ncoercion carr: ∀M:pre_magma. Type ≝ carr on _M: pre_magma to Type.
-ndefinition op ≝
- λM: pre_magma. match M return λM:pre_magma. M → M → M with [ mk_pre_magma _ op ⇒ op ].
 
-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
  }.
-(* this is a projection *)
-ndefinition mcarr ≝ λA.λM: magma A. match M with [ mk_magma mcarr _ ⇒ mcarr ].
-ncoercion mcarr: ∀A.∀M: magma A. Ω \sup A ≝ mcarr
- on _M: magma ? to Ω \sup ?.
-ndefinition op_closed ≝
- λA.λM: magma A.
-  match M return λM: magma A.∀x,y. x ∈ M → y ∈ M → op ? x y ∈ M with
-   [ mk_magma _ opc ⇒ opc ].
 
-nrecord pre_magma_morphism (A,B: pre_magma) : Type ≝
- { mmcarr: A → B;
-   mmprop: ∀x,y. mmcarr (op ? x y) = op ? (mmcarr x) (mmcarr y)
+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)
  }.
-(* this is a projection *)
-ndefinition mmcarr ≝
- λA,B.λf: pre_magma_morphism A B. match f with [ mk_pre_magma_morphism f _ ⇒ f ].
-ncoercion mmcarr: ∀A,B.∀M: pre_magma_morphism A B. A → B ≝ mmcarr
- on _M: pre_magma_morphism ? ? to ∀_.?.
-ndefinition mmprop ≝
- λA,B,M.
-  match M return λM:pre_magma_morphism A B.∀x,y. M (op ? x y) = op ? (M x) (M y) with
-   [ mk_pre_magma_morphism _ p ⇒ p ].
 
 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
  }.
-(* 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 ].
-ncoercion mmmcarr : ∀A,B,Ma,Mb.∀f: magma_morphism A B Ma Mb. pre_magma_morphism A B
- ≝ mmmcarr
- on _f: magma_morphism ???? to pre_magma_morphism ??.
-ndefinition mmclosed ≝
- λA,B,Ma,Mb.λf: magma_morphism A B Ma Mb.
-  match f return λf: magma_morphism A B Ma Mb.∀x. x ∈ Ma → f x ∈ Mb with
-   [ mk_magma_morphism _ p ⇒ p ].
- 
-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; #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