--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_pairs.ma".
+
+(* full_subset e' una coercion che non mette piu' *)
+record concrete_space : Type1 ≝
+ { bp:> BP;
+ converges: ∀a: concr bp.∀U,V: form bp. a ⊩ U → a ⊩ V → a ⊩ (U ↓ V);
+ all_covered: ∀x: concr bp. x ⊩ full_subset (form bp)
+ }.
+
+definition bp': concrete_space → basic_pair ≝ λc.bp c.
+coercion bp'.
+
+definition bp'': concrete_space → objs1 BP ≝ λc.bp c.
+coercion bp''.
+
+record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
+ { rp:> arrows1 ? CS1 CS2;
+ respects_converges:
+ ∀b,c.
+ minus_image ?? rp \sub\c (BPextS CS2 (b ↓ c)) =
+ BPextS CS1 ((minus_image ?? rp \sub\f b) ↓ (minus_image ?? rp \sub\f c));
+ respects_all_covered:
+ minus_image ?? rp\sub\c (BPextS CS2 (full_subset (form CS2))) = BPextS CS1 (full_subset (form CS1))
+ }.
+(*
+definition rp' : ∀CS1,CS2. convergent_relation_pair CS1 CS2 → relation_pair CS1 CS2 ≝
+ λCS1,CS2,c. rp CS1 CS2 c.
+
+coercion rp'.
+
+definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid1.
+ intros;
+ constructor 1;
+ [ apply (convergent_relation_pair c c1)
+ | constructor 1;
+ [ intros;
+ apply (relation_pair_equality c c1 c2 c3);
+ | intros 1; apply refl1;
+ | intros 2; apply sym1;
+ | intros 3; apply trans1]]
+qed.
+
+definition rp'': ∀CS1,CS2.convergent_relation_space_setoid CS1 CS2 → arrows1 BP CS1 CS2 ≝
+ λCS1,CS2,c.rp ?? c.
+
+coercion rp''.
+
+definition convergent_relation_space_composition:
+ ∀o1,o2,o3: concrete_space.
+ binary_morphism1
+ (convergent_relation_space_setoid o1 o2)
+ (convergent_relation_space_setoid o2 o3)
+ (convergent_relation_space_setoid o1 o3).
+ intros; constructor 1;
+ [ intros; whd in c c1 ⊢ %;
+ constructor 1;
+ [ apply (fun1 ??? (comp1 BP ???)); [apply (bp o2) |*: apply rp; assumption]
+ | intros;
+ change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
+ change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? (? ? ? (? ? ? %) ?) ?)))
+ with (c1 \sub \f ∘ c \sub \f);
+ change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? ? (? ? ? (? ? ? %) ?))))
+ with (c1 \sub \f ∘ c \sub \f);
+ apply (.= (extS_com ??????));
+ apply (.= (†(respects_converges ?????)));
+ apply (.= (respects_converges ?????));
+ apply (.= (†(((extS_com ??????) \sup -1)‡(extS_com ??????)\sup -1)));
+ apply refl1;
+ | change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
+ apply (.= (extS_com ??????));
+ apply (.= (†(respects_all_covered ???)));
+ apply (.= respects_all_covered ???);
+ apply refl1]
+ | intros;
+ change with (b ∘ a = b' ∘ a');
+ change in H with (rp'' ?? a = rp'' ?? a');
+ change in H1 with (rp'' ?? b = rp ?? b');
+ apply (.= (H‡H1));
+ apply refl1]
+qed.
+
+definition CSPA: category1.
+ constructor 1;
+ [ apply concrete_space
+ | apply convergent_relation_space_setoid
+ | intro; constructor 1;
+ [ apply id1
+ | intros;
+ unfold id; simplify;
+ apply (.= (equalset_extS_id_X_X ??));
+ apply (.= (†((equalset_extS_id_X_X ??)\sup -1‡
+ (equalset_extS_id_X_X ??)\sup -1)));
+ apply refl1;
+ | apply (.= (equalset_extS_id_X_X ??));
+ apply refl1]
+ | apply convergent_relation_space_composition
+ | intros; simplify;
+ change with (a34 ∘ (a23 ∘ a12) = (a34 ∘ a23) ∘ a12);
+ apply (.= ASSOC1);
+ apply refl1
+ | intros; simplify;
+ change with (a ∘ id1 ? o1 = a);
+ apply (.= id_neutral_right1 ????);
+ apply refl1
+ | intros; simplify;
+ change with (id1 ? o2 ∘ a = a);
+ apply (.= id_neutral_left1 ????);
+ apply refl1]
+qed.
+*)
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