Coercions used here and there.

diff --git a/helm/software/matita/nlibrary/algebra/magmas.ma b/helm/software/matita/nlibrary/algebra/magmas.ma
index dd6f9ae48f77ee8452b08fbb59226f9a7cd4634f..e49d8ea82545549e7b9c3818f1ba029879ede330 100644 (file)
--- a/helm/software/matita/nlibrary/algebra/magmas.ma
+++ b/helm/software/matita/nlibrary/algebra/magmas.ma
@@ -19,44 +19,56 @@ nrecord pre_magma : Type[1] ≝
    op: carr → carr → carr
  }.
 (* this is a projection *)
-ndefinition carr ≝ λM: pre_magma. match M with [ mk_pre_magma carr _ ⇒ carr ].
+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. carr M → carr M → carr M with [ mk_pre_magma _ op ⇒ op ].
-(* ncoercion carr. *)
+ 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 (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 ].
+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.∀x,y. x ∈ mcarr ? M → y ∈ mcarr ? M → op A x y ∈ mcarr ? M with
+  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: carr A → carr B;
+ { mmcarr: 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 ∀_.?.
 
 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
+   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 mmcarr_mmmcarr ≝
+ λA,B,Ma,Mb.λf: magma_morphism A B Ma Mb. mmcarr ?? (mmmcarr ???? f).
+ncoercion mmcarr_mmmcarr : ∀A,B,Ma,Mb.∀f: magma_morphism A B Ma Mb. A → B ≝ mmcarr_mmmcarr
+ on _f: magma_morphism ???? to ∀_.?.
 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
+  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. mcarr ? M1 ⊆ 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}.