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_type : Type[1] ≝
19 op: binary_morphism mtcarr mtcarr mtcarr
22 nrecord magma (A: magma_type) : Type[1] ≝
23 { mcarr:> qpowerclass A;
24 op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
27 ncoercion mcarr' : ∀A. ∀M: magma A. carr1 (qpowerclass_setoid (mtcarr A))
28 ≝ λA.λM: magma A.mcarr ? M
29 on _M: magma ? to carr1 (qpowerclass_setoid (mtcarr ?)).
31 nrecord magma_morphism_type (A,B: magma_type) : Type[0] ≝
32 { mmcarr:> unary_morphism A B;
33 mmprop: ∀x,y:A. mmcarr (op ? x y) = op … (mmcarr x) (mmcarr y)
36 nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type[0] ≝
37 { mmmcarr:> magma_morphism_type A B;
38 mmclosed: ∀x:A. x ∈ mcarr ? Ma → mmmcarr x ∈ mcarr ? Mb
43 ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma B.
46 [ napply (image … f Ma)
47 | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
51 [ napply op_closed; nassumption
54 napply (mmprop … f)]##]
57 ndefinition mm_counter_image:
58 ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma A.
61 [ napply (counter_image … f Mb)
62 | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
66 [ napply op_closed; nassumption
69 napply (mmprop … f)]##]
73 ndefinition m_intersect: ∀A. magma A → magma A → magma A.
76 [ napply (intersect_ok ? M1 M2)
77 | #x; #y; nwhd in ⊢ (% → % → %); *; #Hx1; #Hx2; *; #Hy1; #Hy2;
78 napply conj; napply op_closed; nassumption ]