topology/igt.ma (???) |-> sets/setoids.ma

[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 22289b3c2c6e486d47c4e68d2fca99b4953aea41..842cfceb93c0aa82f50878294937479361911714 100644 (file)
--- a/helm/software/matita/nlibrary/algebra/magmas.ma
+++ b/helm/software/matita/nlibrary/algebra/magmas.ma
@@ -14,36 +14,25 @@
 
 include "sets/sets.ma".
 
-nrecord pre_magma : Type[1] ≝
- { carr: Type;
+nrecord magma_type : Type[1] ≝
+ { carr:> Type;
    op: carr → carr → carr
  }.
-ncoercion carr: ∀M:pre_magma. Type ≝ carr on _M: pre_magma to Type.
 
-nrecord magma (A: pre_magma) : Type[1] ≝
- { mcarr: Ω \sup A;
+nrecord magma (A: magma_type) : Type[1] ≝
+ { mcarr:> Ω \sup A;
    op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
  }.
-ncoercion mcarr: ∀A.∀M: magma A. Ω \sup A ≝ mcarr
- on _M: magma ? to Ω \sup ?.
 
-nrecord pre_magma_morphism (A,B: pre_magma) : Type ≝
- { mmcarr: A → B;
+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)
  }.
-ncoercion mmcarr: ∀A,B.∀M: pre_magma_morphism A B. A → B ≝ mmcarr
- on _M: pre_magma_morphism ? ? to ∀_.?.
 
 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
  }.
-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 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}.
@@ -79,4 +68,12 @@ ndefinition mm_counter_image:
          | 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