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 "basic_pairs.ma".
17 (* full_subset e' una coercion che non mette piu' *)
18 record concrete_space : Type1 ≝
20 converges: ∀a: concr bp.∀U,V: form bp. a ⊩ U → a ⊩ V → a ⊩ (U ↓ V);
21 all_covered: ∀x: concr bp. x ⊩ full_subset (form bp)
24 record convergent_relation_pair (CS1,CS2: concrete_space) : Type1 ≝
25 { rp:> arrows1 ? CS1 CS2;
28 minus_image ?? rp \sub\c (BPextS CS2 (b ↓ c)) =
29 BPextS CS1 ((minus_image ?? rp \sub\f b) ↓ (minus_image ?? rp \sub\f c));
31 minus_image ?? rp\sub\c (BPextS CS2 (full_subset (form CS2))) = BPextS CS1 (full_subset (form CS1))
34 definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid1.
37 [ apply (convergent_relation_pair c c1)
40 apply (relation_pair_equality c c1 c2 c3);
41 | intros 1; apply refl1;
42 | intros 2; apply sym1;
43 | intros 3; apply trans1]]
46 definition convergent_relation_space_composition:
47 ∀o1,o2,o3: concrete_space.
49 (convergent_relation_space_setoid o1 o2)
50 (convergent_relation_space_setoid o2 o3)
51 (convergent_relation_space_setoid o1 o3).
52 intros; constructor 1;
53 [ intros; whd in c c1 ⊢ %;
55 [ apply (fun1 ??? (comp1 BP ???)); [apply (bp o2) |*: apply rp; assumption]
57 change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
58 change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? (? ? ? (? ? ? %) ?) ?)))
59 with (c1 \sub \f ∘ c \sub \f);
60 change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? ? (? ? ? (? ? ? %) ?))))
61 with (c1 \sub \f ∘ c \sub \f);
62 apply (.= (extS_com ??????));
63 apply (.= (†(respects_converges ?????)));
64 apply (.= (respects_converges ?????));
65 apply (.= (†(((extS_com ??????) \sup -1)‡(extS_com ??????)\sup -1)));
67 | change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
68 apply (.= (extS_com ??????));
69 apply (.= (†(respects_all_covered ???)));
70 apply (.= respects_all_covered ???);
73 change with (b ∘ a = b' ∘ a');
74 change in H with (rp'' ?? a = rp'' ?? a');
75 change in H1 with (rp'' ?? b = rp ?? b');
80 definition CSPA: category1.
82 [ apply concrete_space
83 | apply convergent_relation_space_setoid
84 | intro; constructor 1;
88 apply (.= (equalset_extS_id_X_X ??));
89 apply (.= (†((equalset_extS_id_X_X ??)\sup -1‡
90 (equalset_extS_id_X_X ??)\sup -1)));
92 | apply (.= (equalset_extS_id_X_X ??));
94 | apply convergent_relation_space_composition
96 change with (a34 ∘ (a23 ∘ a12) = (a34 ∘ a23) ∘ a12);
100 change with (a ∘ id1 ? o1 = a);
101 apply (.= id_neutral_right1 ????);
104 change with (id1 ? o2 ∘ a = a);
105 apply (.= id_neutral_left1 ????);