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[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 3424944bedea48628147f3165766185e2cd531b8..0d4d7d6b10e06f689f271e612c9f3d1bc106a5b7 100644 (file)
--- a/helm/software/matita/nlibrary/algebra/magmas.ma
+++ b/helm/software/matita/nlibrary/algebra/magmas.ma
@@ -14,8 +14,62 @@
 
 include "sets/sets.ma".
 
-nrecord magma (A: Type) : Type[1] ≝
- { 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 pre_magma : Type[1] ≝
+ { 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);
+   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;
+   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
+ }.
+(* 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.
+ 
+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 (ex_intro ????)
+     [ napply (op ? x0 y0) 
+     | nelim daemon ]##]
+nqed.
\ No newline at end of file