1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "sets/sets.ma".
17 nrecord magma (A: Type) : Type[1] ≝
20 op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op x y ∈ mcarr
22 (* this is a projection *)
23 ndefinition mcarr ≝ λA.λM: magma A. match M with [ mk_magma mcarr _ _ ⇒ mcarr ].
24 ndefinition op ≝ λA.λM: magma A. match M with [ mk_magma _ op _ ⇒ op ].
27 nrecord magma_morphism (A,B: Type) (Ma: magma A) (Mb: magma B) : Type ≝
29 mmclosed: ∀x. x ∈ mcarr ? Ma → mmcarr x ∈ mcarr ? Mb;
30 (* need a canonical structure in next line? *)
31 mmprop: ∀x,y:A. x ∈ mcarr ? Ma → y ∈ mcarr ? Ma → mmcarr (op ? Ma x y) = op B Mb (mmcarr x) (mmcarr y)
33 (* this is a projection *)
35 λA,B,Ma,Mb.λf: magma_morphism A B Ma Mb. match f with [ mk_magma_morphism f _ _ ⇒ f ].
37 ndefinition sub_magma ≝
38 λA.λM1,M2: magma A. ∀x. x ∈ mcarr ? M1 → x ∈ mcarr ? M2.
40 ndefinition image: ∀A,B. (A → B) → Ω \sup A → Ω \sup B ≝
41 λA,B,f,Sa. {y | ∃x. x ∈ Sa ∧ f x = y}.
46 ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism ?? Ma Mb → magma B.
48 napply (mk_magma ????)
49 [ napply (image ?? (mmcarr ???? f) (mcarr ? Ma))
51 | #x; #y; *; #x0; #Hx0; *; #y0; #Hy0; nwhd;
52 napply (ex_intro ????)
53 [ napply (op ? Ma x0 y0) (* BAD HERE! need a canonical structure? *)