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 definition bp': concrete_space → basic_pair ≝ λc.bp c.
27 definition bp'': concrete_space → objs1 BP ≝ λc.bp c.
30 record convergent_relation_pair (CS1,CS2: concrete_space) : Type1 ≝
31 { rp:> arrows1 ? CS1 CS2;
34 minus_image ?? rp \sub\c (BPextS CS2 (b ↓ c)) =
35 BPextS CS1 ((minus_image ?? rp \sub\f b) ↓ (minus_image ?? rp \sub\f c));
37 minus_image ?? rp\sub\c (BPextS CS2 (full_subset (form CS2))) = BPextS CS1 (full_subset (form CS1))
40 definition rp' : ∀CS1,CS2. convergent_relation_pair CS1 CS2 → relation_pair CS1 CS2 ≝
41 λCS1,CS2,c. rp CS1 CS2 c.
45 definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid1.
48 [ apply (convergent_relation_pair c c1)
51 apply (relation_pair_equality c c1 c2 c3);
52 | intros 1; apply refl1;
53 | intros 2; apply sym1;
54 | intros 3; apply trans1]]
57 definition rp'': ∀CS1,CS2.convergent_relation_space_setoid CS1 CS2 → arrows1 BP CS1 CS2 ≝
62 definition convergent_relation_space_composition:
63 ∀o1,o2,o3: concrete_space.
65 (convergent_relation_space_setoid o1 o2)
66 (convergent_relation_space_setoid o2 o3)
67 (convergent_relation_space_setoid o1 o3).
68 intros; constructor 1;
69 [ intros; whd in c c1 ⊢ %;
71 [ apply (fun1 ??? (comp1 BP ???)); [apply (bp o2) |*: apply rp; assumption]
73 change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
74 change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? (? ? ? (? ? ? %) ?) ?)))
75 with (c1 \sub \f ∘ c \sub \f);
76 change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? ? (? ? ? (? ? ? %) ?))))
77 with (c1 \sub \f ∘ c \sub \f);
78 apply (.= (extS_com ??????));
79 apply (.= (†(respects_converges ?????)));
80 apply (.= (respects_converges ?????));
81 apply (.= (†(((extS_com ??????) \sup -1)‡(extS_com ??????)\sup -1)));
83 | change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
84 apply (.= (extS_com ??????));
85 apply (.= (†(respects_all_covered ???)));
86 apply (.= respects_all_covered ???);
89 change with (b ∘ a = b' ∘ a');
90 change in H with (rp'' ?? a = rp'' ?? a');
91 change in H1 with (rp'' ?? b = rp ?? b');
96 definition CSPA: category1.
98 [ apply concrete_space
99 | apply convergent_relation_space_setoid
100 | intro; constructor 1;
104 apply (.= (equalset_extS_id_X_X ??));
105 apply (.= (†((equalset_extS_id_X_X ??)\sup -1‡
106 (equalset_extS_id_X_X ??)\sup -1)));
108 | apply (.= (equalset_extS_id_X_X ??));
110 | apply convergent_relation_space_composition
112 change with (a34 ∘ (a23 ∘ a12) = (a34 ∘ a23) ∘ a12);
116 change with (a ∘ id1 ? o1 = a);
117 apply (.= id_neutral_right1 ????);
120 change with (id1 ? o2 ∘ a = a);
121 apply (.= id_neutral_left1 ????);