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 ncoercion carr: ∀M:pre_magma. Type ≝ carr on _M: pre_magma to Type.
23 nrecord magma (A: pre_magma) : Type[1] ≝
25 op_closed: ∀x,y. x ∈ mcarr → y ∈ mcarr → op A x y ∈ mcarr
27 ncoercion mcarr: ∀A.∀M: magma A. Ω \sup A ≝ mcarr
28 on _M: magma ? to Ω \sup ?.
30 nrecord pre_magma_morphism (A,B: pre_magma) : Type ≝
32 mmprop: ∀x,y. mmcarr (op ? x y) = op ? (mmcarr x) (mmcarr y)
34 ncoercion mmcarr: ∀A,B.∀M: pre_magma_morphism A B. A → B ≝ mmcarr
35 on _M: pre_magma_morphism ? ? to ∀_.?.
37 nrecord magma_morphism (A) (B) (Ma: magma A) (Mb: magma B) : Type ≝
38 { mmmcarr: pre_magma_morphism A B;
39 mmclosed: ∀x. x ∈ Ma → mmmcarr x ∈ Mb
41 ncoercion mmmcarr : ∀A,B,Ma,Mb.∀f: magma_morphism A B Ma Mb. pre_magma_morphism A B
43 on _f: magma_morphism ???? to pre_magma_morphism ??.
45 ndefinition sub_magma ≝
46 λA.λM1,M2: magma A. M1 ⊆ M2.
48 ndefinition image: ∀A,B. (A → B) → Ω \sup A → Ω \sup B ≝
49 λA,B,f,Sa. {y | ∃x. x ∈ Sa ∧ f x = y}.
52 ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism ?? Ma Mb → magma B.
55 [ napply (image ?? f Ma)
56 | #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
57 napply (ex_intro ????)
60 [ napply (op_closed ??????); nassumption
63 napply (mmprop ?? f ??)]##]
66 ndefinition counter_image: ∀A,B. (A → B) → Ω \sup B → Ω \sup A ≝
67 λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
69 ndefinition mm_counter_image:
70 ∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism ?? Ma Mb → magma A.
73 [ napply (counter_image ?? f Mb)
74 | #x; #y; nwhd in ⊢ (% → % → ?); *; #y0; *; #Hy0; #H