∀P,Q. arrows1 REL P Q → binary_relation_setoid P Q ≝ λP,Q,x.x.
coercion binary_relation_setoid_of_arrow1_REL.
+
+notation > "B ⇒_\r1 C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
+notation "B ⇒\sub (\r 1) C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
+interpretation "'arrows1_SET" 'arrows1_REL A B = (arrows1 REL A B).
+
+
definition full_subset: ∀s:REL. Ω^s.
apply (λs.{x | True});
intros; simplify; split; intro; assumption.
coercion full_subset.
-alias symbol "arrows1_SET" (instance 2) = "'arrows1_SET low".
-definition comprehension: ∀b:REL. (b ⇒_1 CPROP) → Ω^b.
+definition comprehension: ∀b:REL. (b ⇒_1. CPROP) → Ω^b.
apply (λb:REL. λP: b ⇒_1 CPROP. {x | P x});
intros; simplify;
apply (.= †e); apply refl1.
interpretation "subset comprehension" 'comprehension s p =
(comprehension s (mk_unary_morphism1 ?? p ?)).
-definition ext: ∀X,S:REL. (arrows1 ? X S) × S ⇒_1 (Ω^X).
- apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 REL X S.λf:S.{x ∈ X | x ♮r f}) ?);
- [ intros; simplify; apply (.= (e‡#)); apply refl1
+definition ext: ∀X,S:REL. (X ⇒_\r1 S) × S ⇒_1 (Ω^X).
+ intros (X S); constructor 1;
+ [ apply (λr:X ⇒_\r1 S.λf:S.{x ∈ X | x ♮r f}); intros; simplify; apply (.= (e‡#)); apply refl1
| intros; simplify; split; intros; simplify;
[ change with (∀x. x ♮a b → x ♮a' b'); intros;
apply (. (#‡e1^-1)); whd in e; apply (if ?? (e ??)); assumption
| change with (∀x. x ♮a' b' → x ♮a b); intros;
apply (. (#‡e1)); whd in e; apply (fi ?? (e ??));assumption]]
qed.
+
(*
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});*)
*)
(* the same as ⋄ for a basic pair *)
-definition image: ∀U,V:REL. (arrows1 ? U V) × Ω^U ⇒_1 Ω^V.
+definition image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
+ [ apply (λr:U ⇒_\r1 V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
intros; simplify; split; intro; cases e1; exists [1,3: apply w]
[ apply (. (#‡e^-1)‡#); assumption
| apply (. (#‡e)‡#); assumption]
qed.
(* the same as □ for a basic pair *)
-definition minus_star_image: ∀U,V:REL. (arrows1 ? U V) × Ω^U ⇒_1 Ω^V.
+definition minus_star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
+ [ apply (λr:U ⇒_\r1 V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
intros; simplify; split; intros; apply f;
[ apply (. #‡e); assumption
| apply (. #‡e ^ -1); assumption]
qed.
(* the same as Rest for a basic pair *)
-definition star_image: ∀U,V:REL. (arrows1 ? U V) × Ω^V ⇒_1 Ω^U.
+definition star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
+ [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
intros; simplify; split; intros; apply f;
[ apply (. e ‡#); assumption
| apply (. e^ -1‡#); assumption]
qed.
(* the same as Ext for a basic pair *)
-definition minus_image: ∀U,V:REL. (arrows1 ? U V) × Ω^V ⇒_1 Ω^U.
+definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
+ [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
exT ? (λy:V.x ♮r y ∧ y ∈ S) });
intros; simplify; split; intro; cases e1; exists [1,3: apply w]
[ apply (. (e ^ -1‡#)‡#); assumption