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 pre_magma : Type[1] ≝
19 op: carr → carr → carr
21 (* this is a projection *)
22 ndefinition carr ≝ λM: pre_magma. match M with [ mk_pre_magma carr _ ⇒ carr ].
24 λM: pre_magma. match M return λM. carr M → carr M → carr M with [ mk_pre_magma _ op ⇒ op ].
27 nrecord magma (A: pre_magma) : Type[1] ≝
28 { mcarr: Ω \sup (carr A);
29 op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
31 (* this is a projection *)
32 ndefinition mcarr ≝ λA.λM: magma A. match M with [ mk_magma mcarr _ ⇒ mcarr ].
33 ndefinition op_closed ≝
35 match M return λM.∀x,y. x ∈ mcarr ? M → y ∈ mcarr ? M → op A x y ∈ mcarr ? M with
36 [ mk_magma _ opc ⇒ opc ].
38 nrecord pre_magma_morphism (A,B: pre_magma) : Type ≝
39 { mmcarr: carr A → carr B;
40 mmprop: ∀x,y. mmcarr (op ? x y) = op ? (mmcarr x) (mmcarr y)
42 (* this is a projection *)
44 λA,B.λf: pre_magma_morphism A B. match f with [ mk_pre_magma_morphism f _ ⇒ f ].
46 nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type ≝
47 { mmmcarr: pre_magma_morphism A B;
48 mmclosed: ∀x. x ∈ mcarr ? Ma → mmcarr ?? mmmcarr x ∈ mcarr ? Mb
50 (* this is a projection *)
52 λA,B,Ma,Mb.λf: magma_morphism A B Ma Mb. match f with [ mk_magma_morphism f _ ⇒ f ].
53 ndefinition mmclosed ≝
54 λA,B,Ma,Mb.λf: magma_morphism A B Ma Mb.
55 match f return λf.∀x. x ∈ mcarr ? Ma → mmcarr ?? (mmmcarr ???? f) x ∈ mcarr ? Mb with
56 [ mk_magma_morphism _ p ⇒ p ].
58 ndefinition sub_magma ≝
59 λA.λM1,M2: magma A. mcarr ? M1 ⊆ mcarr ? M2.
61 ndefinition image: ∀A,B. (A → B) → Ω \sup A → Ω \sup B ≝
62 λA,B,f,Sa. {y | ∃x. x ∈ Sa ∧ f x = y}.
67 ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism ?? Ma Mb → magma B.
70 [ napply (image ?? (mmcarr ?? (mmmcarr ???? f)) (mcarr ? Ma))
71 | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
72 napply (ex_intro ????)
75 [ napply (op_closed ??????); nassumption