include "formal_topology/basic_pairs.ma".
-interpretation "REL carrier" 'card c = (carrier c).
-
-definition comprehension: ∀b:REL. (b → CProp) → Ω \sup |b| ≝
- λb:REL.λP.{x | ∃p: x ∈ b. P (mk_ssigma ?? x p)}.
+definition comprehension: ∀b:REL. (b ⇒ CPROP) → Ω \sup b.
+ apply (λb:REL. λP: b ⇒ CPROP. {x | x ∈ b ∧ P x});
+ intros; simplify; apply (.= (H‡#)‡(†H)); apply refl1.
+qed.
interpretation "subset comprehension" 'comprehension s p =
- (comprehension s p).
+ (comprehension s (mk_unary_morphism __ p _)).
+
+definition ext: ∀X,S:REL. ∀r: arrows1 ? X S. S ⇒ Ω \sup X.
+ apply (λX,S,r.mk_unary_morphism ?? (λf.{x ∈ X | x ♮r f}) ?);
+ [ intros; simplify; apply (.= (H‡#)); apply refl1
+ | intros; simplify; split; intros; simplify; intros;
+ [ apply (. #‡(#‡H)); assumption
+ | apply (. #‡(#‡H\sup -1)); assumption]]
+qed.
+
+definition BPext: ∀o: basic_pair. form o ⇒ Ω \sup (concr o) ≝ λo.ext ? ? (rel o).
+
+definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup X.
+ (* ∃ is not yet a morphism apply (λX,S,r,F.{x ∈ X | ∃a. a ∈ F ∧ x ♮r a});*)
+ intros (X S r); constructor 1;
+ [ intro F; constructor 1; constructor 1;
+ [ apply (λx. x ∈ X ∧ ∃a:S. a ∈ F ∧ x ♮r a);
+ | intros; split; intro; cases f (H1 H2); clear f; split;
+ [ apply (. (H‡#)); assumption
+ |3: apply (. (H\sup -1‡#)); assumption
+ |2,4: cases H2 (w H3); exists; [1,3: apply w]
+ [ apply (. (#‡(H‡#))); assumption
+ | apply (. (#‡(H \sup -1‡#))); assumption]]]
+ | intros; split; simplify; intros; cases f; cases H1; split;
+ [1,3: assumption
+ |2,4: exists; [1,3: apply w]
+ [ apply (. (#‡H)‡#); assumption
+ | apply (. (#‡H\sup -1)‡#); assumption]]]
+qed.
+
+definition BPextS: ∀o: basic_pair. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
+ λo.extS ?? (rel o).
+
+definition fintersects: ∀o: basic_pair. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
+ intros (o); constructor 1;
+ [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
+ intros; simplify; apply (.= (†H)‡#); apply refl1
+ | intros; split; simplify; intros;
+ [ apply (. #‡((†H)‡(†H1))); assumption
+ | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+qed.
+
+interpretation "fintersects" 'fintersects U V = (fun1 ___ (fintersects _) U V).
+
+definition fintersectsS:
+ ∀o:basic_pair. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
+ intros (o); constructor 1;
+ [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
+ intros; simplify; apply (.= (†H)‡#); apply refl1
+ | intros; split; simplify; intros;
+ [ apply (. #‡((†H)‡(†H1))); assumption
+ | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+qed.
+
+interpretation "fintersectsS" 'fintersects U V = (fun1 ___ (fintersectsS _) U V).
+
+definition relS: ∀o: basic_pair. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
+ intros (o); constructor 1;
+ [ apply (λx:concr o.λS: Ω \sup (form o).∃y: form o.y ∈ S ∧ x ⊩ y);
+ | intros; split; intros; cases H2; exists [1,3: apply w]
+ [ apply (. (#‡H1)‡(H‡#)); assumption
+ | apply (. (#‡H1 \sup -1)‡(H \sup -1‡#)); assumption]]
+qed.
+
+interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr _) __ (relS _) x y).
+interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ___ (relS _)).
+
+record concrete_space : Type ≝
+ { bp:> basic_pair;
+ converges: ∀a: concr bp.∀U,V: form bp. a ⊩ U → a ⊩ V → a ⊩ (U ↓ V);
+ all_covered: ∀x: concr bp. x ⊩ form bp
+ }.
+
+(*
+record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
+ { rp:> relation_pair CS1 CS2;
+ respects_converges:
+ ∀b,c.
+ extS ?? rp \sub\c (extS ?? (rel CS2) (b ↓ c)) =
+ extS ?? (rel CS1) ((extS ?? rp \sub\f b) ↓ (extS ?? rp \sub\f c));
+ respects_all_covered:
+ extS ?? rp\sub\c (extS ?? (rel CS2) (form CS2)) =
+ extS ?? (rel CS1) (form CS1)
+ }.
+
+definition convergent_relation_space_setoid: concrete_space → concrete_space → setoid.
+ intros;
+ constructor 1;
+ [ apply (convergent_relation_pair c c1)
+ | constructor 1;
+ [ intros;
+ apply (relation_pair_equality c c1 c2 c3);
+ | intros 1; apply refl;
+ | intros 2; apply sym;
+ | intros 3; apply trans]]
+qed.
+
+lemma equalset_extS_id_X_X: ∀o:REL.∀X.extS ?? (id ? o) X = X.
+ intros;
+ unfold extS;
+ split;
+ [ intros 2;
+ cases m; clear m;
+ cases H; clear H;
+ cases H1; clear H1;
+ whd in H;
+ apply (eq_elim_r'' ????? H);
+ assumption
+ | intros 2;
+ constructor 1;
+ [ whd; exact I
+ | exists; [ apply a ]
+ split;
+ [ assumption
+ | whd; constructor 1]]]
+qed.
-definition ext: ∀o: basic_pair. (form o) → Ω \sup |(concr o)| ≝
- λo,f.{x ∈ (concr o) | x ♮(rel o) f}.
+definition CSPA: category.
+ constructor 1;
+ [ apply concrete_space
+ | apply convergent_relation_space_setoid
+ | intro; constructor 1;
+ [ apply id
+ | intros;
+ unfold id; simplify;
+ apply (.= (equalset_extS_id_X_X ??));
-definition downarrow ≝
- λo: basic_pair.λa,b: form o.
- {c | ext ? c ⊆ ext ? a ∩ ext ? b }.
\ No newline at end of file
+ |
+ ]
+ |*)
\ No newline at end of file
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