include "formal_topology/basic_pairs.ma".
-interpretation "REL carrier" 'card c = (carrier c).
+record concrete_space : Type ≝
+ { bp:> BP;
+ converges: ∀a: concr bp.∀U,V: form bp. a ⊩ U → a ⊩ V → a ⊩ (U ↓ V);
+ all_covered: ∀x: concr bp. x ⊩ form bp
+ }.
-definition comprehension: ∀b:REL. (b → CProp) → Ω \sup |b| ≝
- λb:REL.λP.{x | ∃p: x ∈ b. P (mk_ssigma ?? x p)}.
+definition bp': concrete_space → basic_pair ≝ λc.bp c.
-interpretation "subset comprehension" 'comprehension s p =
- (comprehension s p).
+coercion bp'.
-definition ext: ∀o: basic_pair. form o → Ω \sup |(concr o)| ≝
- λo,f.{x ∈ (concr o) | x ♮(rel o) f}.
+record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
+ { rp:> arrows1 ? CS1 CS2;
+ respects_converges:
+ ∀b,c.
+ extS ?? rp \sub\c (BPextS CS2 (b ↓ c)) =
+ BPextS CS1 ((extS ?? rp \sub\f b) ↓ (extS ?? rp \sub\f c));
+ respects_all_covered:
+ extS ?? rp\sub\c (BPextS CS2 (form CS2)) = BPextS CS1 (form CS1)
+ }.
-definition fintersects ≝
- λo: basic_pair.λa,b: form o.
- {c | ext ? c ⊆ ext ? a ∩ ext ? b }.
+definition rp' : ∀CS1,CS2. convergent_relation_pair CS1 CS2 → relation_pair CS1 CS2 ≝
+ λCS1,CS2,c. rp CS1 CS2 c.
+
+coercion rp'.
-interpretation "fintersects" 'fintersects U V = (fintersects _ U V).
+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 relS: ∀o: basic_pair. concr o → Ω \sup (form o) → CProp ≝
- λo,x,S. ∃y. y ∈ S ∧ x ⊩ y.
+definition rp'': ∀CS1,CS2.convergent_relation_space_setoid CS1 CS2 → arrows1 BP CS1 CS2 ≝
+ λCS1,CS2,c.rp ?? c.
-interpretation "basic pair relation for subsets" 'Vdash2 x y = (relS _ x y).
-interpretation "basic pair relation for subsets (non applied)" 'Vdash = (relS _).
+coercion rp''.
-definition convergence ≝
- λo: basic_pair.∀a: concr o.∀U,V: form o. a ⊩ U → a ⊩ V → a ⊩ (U ↓ V).
\ No newline at end of file
+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.