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:> powerset_setoid1 A;
24 op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
26 (* le coercion non vanno; sospetto setoid1_of_setoid *)
27 nrecord magma_morphism_type (A,B: magma_type) : Type[0] ≝
28 { mmcarr:> unary_morphism A B;
29 mmprop: ∀x,y:carr A. mmcarr (op ? x y) = op … (mmcarr x) (mmcarr y)
31 (* le coercion non vanno *)
32 nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type[0] ≝
33 { mmmcarr:> magma_morphism_type A B;
34 mmclosed: ∀x:carr A. x ∈ mcarr ? Ma → mmmcarr x ∈ mcarr ? Mb
37 (* qui non funziona una cippa *)
38 ndefinition image: ∀A,B. (carr A → carr B) → Ω \sup A → Ω \sup B ≝
39 λA,B:setoid.λf:carr A → carr B.λSa:Ω \sup A.
40 {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) ? ?(*(f x) y*)}.
41 ##[##2: napply (f x); ##|##3: napply y]
42 #a; #a'; #H; nwhd; nnormalize; (* per togliere setoid1_of_setoid *) napply (mk_iff ????);
43 *; #x; #Hx; napply (ex_intro … x)
44 [ napply (. (#‡(#‡#)));
47 ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma B.
50 [ napply (image … f Ma)
51 | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
55 [ napply op_closed; nassumption
58 napply (mmprop … f)]##]
61 ndefinition counter_image: ∀A,B. (A → B) → Ω \sup B → Ω \sup A ≝
62 λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
64 ndefinition mm_counter_image:
65 ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma A.
68 [ napply (counter_image … f Mb)
69 | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
73 [ napply op_closed; nassumption
76 napply (mmprop … f)]##]
79 ndefinition m_intersect: ∀A. magma A → magma A → magma A.
83 | #x; #y; nwhd in ⊢ (% → % → %); *; #Hx1; #Hx2; *; #Hy1; #Hy2;
84 napply conj; napply op_closed; nassumption ]