No more hand-made coercions.

[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 28b3795c88098743ba2648392ff77ef04c1244b1..1c829997aaf14724a02f5a39c2e22e411d06edf5 100644 (file)
--- a/helm/software/matita/nlibrary/algebra/magmas.ma
+++ b/helm/software/matita/nlibrary/algebra/magmas.ma
@@ -15,62 +15,60 @@
 include "sets/sets.ma".
 
 nrecord pre_magma : Type[1] ≝
- { carr: Type;
+ { carr:> Type;
    op: carr → carr → carr
  }.
-(* this is a projection *)
-ndefinition carr ≝ λM: pre_magma. match M with [ mk_pre_magma carr _ ⇒ carr ].
-ndefinition op ≝
- λM: pre_magma. match M return λM. carr M → carr M → carr M with [ mk_pre_magma _ op ⇒ op ].
-(* ncoercion carr. *)
 
 nrecord magma (A: pre_magma) : Type[1] ≝
- { mcarr: Ω \sup (carr A);
+ { mcarr:> Ω \sup 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 ].
-ndefinition op_closed ≝
- λA.λM: magma A.
-  match M return λM.∀x,y. x ∈ mcarr ? M → y ∈ mcarr ? M → op A x y ∈ mcarr ? M with
-   [ mk_magma _ opc ⇒ opc ].
 
 nrecord pre_magma_morphism (A,B: pre_magma) : Type ≝
- { mmcarr: carr A → carr B;
+ { 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 ].
 
 nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type ≝
- { mmmcarr: pre_magma_morphism A B;
-   mmclosed: ∀x. x ∈ mcarr ? Ma → mmcarr ?? mmmcarr x ∈ mcarr ? Mb
+ { mmmcarr:> pre_magma_morphism 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 ].
-ndefinition mmclosed ≝
- λA,B,Ma,Mb.λf: magma_morphism A B Ma Mb.
-  match f return λf.∀x. x ∈ mcarr ? Ma → mmcarr ?? (mmmcarr ???? f) x ∈ mcarr ? Mb with
-   [ mk_magma_morphism _ p ⇒ p ].
  
 ndefinition sub_magma ≝
- λA.λM1,M2: magma A. ∀x. x ∈ mcarr ? M1 → x ∈ mcarr ? M2.
+ 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}.
 
-naxiom daemon: False.
-
 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 ?? (mmcarr ?? (mmmcarr ???? f)) (mcarr ? Ma)) 
-  | #x; #y; *; #x0; #Hx0; *; #y0; #Hy0; nwhd;
+  [ napply (image ?? f Ma)
+  | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
     napply (ex_intro ????)
      [ napply (op ? x0 y0) 
-     | napply (conj ????);
-       nelim daemon ]##]
+     | 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.
\ No newline at end of file