-definition REL: category.
- constructor 1;
- [ apply (ΣA:Type.Ω \sup A)
- | intros; apply (binary_relation_setoid ?? (s_proof ?? s) (s_proof ?? s1))
- | intros; constructor 1; intros; apply (s=s1)
- | intros; constructor 1;
- [ apply composition
- | apply composition_morphism
- ]
- | intros; unfold mk_binary_morphism; simplify;
- apply associative_composition
- |6,7: intros 5; simplify; split; intro;
- [1,3: cases H; clear H; cases H1; clear H1;
- [ rewrite > H | rewrite < H2 ]
- assumption
- |*: exists; try assumption; split; first [ reflexivity | assumption ]]]
+(* the same as Ext for a basic pair *)
+definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^V ⇒_2 Ω^U).
+ intros; constructor 1;
+ [ intro r; constructor 1;
+ [ apply (λS: Ω^V. {x | ∃y:V. x ♮r y ∧ y ∈ S }).
+ intros; simplify; split; intros; cases e1; cases x; exists; [1,3: apply w]
+ split; try assumption; [ apply (. (e^-1‡#)); | apply (. (e‡#));] assumption;
+ | intros; simplify; split; simplify; intros; cases e1; cases x;
+ exists [1,3: apply w] split; try assumption;
+ [ apply (. (#‡e^-1)); | apply (. (#‡e));] assumption]
+ | intros; intro; simplify; split; simplify; intros; cases e1; exists [1,3: apply w]
+ cases x; split; try assumption;
+ [ apply (. e^-1 a2 w); | apply (. e a2 w)] assumption;]
+qed.
+
+definition foo : ∀o1,o2:REL.carr1 (o1 ⇒_\r1 o2) → carr2 (setoid2_of_setoid1 (o1 ⇒_\r1 o2)) ≝ λo1,o2,x.x.
+
+interpretation "relation f⎻*" 'OR_f_minus_star r = (fun12 ?? (minus_star_image ? ?) (foo ?? r)).
+interpretation "relation f⎻" 'OR_f_minus r = (fun12 ?? (minus_image ? ?) (foo ?? r)).
+interpretation "relation f*" 'OR_f_star r = (fun12 ?? (star_image ? ?) (foo ?? r)).
+
+definition image_coercion: ∀U,V:REL. (U ⇒_\r1 V) → Ω^U ⇒_2 Ω^V.
+intros (U V r Us); apply (image U V r); qed.
+coercion image_coercion.
+
+(* minus_image is the same as ext *)
+
+theorem image_id: ∀o. (id1 REL o : carr2 (Ω^o ⇒_2 Ω^o)) =_1 (id2 SET1 Ω^o).
+ intros; unfold image_coercion; unfold image; simplify;
+ whd in match (?:carr2 ?);
+ intro U; simplify; split; simplify; intros;
+ [ change with (a ∈ U);
+ cases e; cases x; change in e1 with (w =_1 a); apply (. e1^-1‡#); assumption
+ | change in f with (a ∈ U);
+ exists; [apply a] split; [ change with (a = a); apply refl1 | assumption]]
+qed.
+
+theorem minus_star_image_id: ∀o:REL.
+ ((id1 REL o)⎻* : carr2 (Ω^o ⇒_2 Ω^o)) =_1 (id2 SET1 Ω^o).
+ intros; unfold minus_star_image; simplify; intro U; simplify;
+ split; simplify; intros;
+ [ change with (a ∈ U); apply f; change with (a=a); apply refl1
+ | change in f1 with (eq1 ? x a); apply (. f1‡#); apply f]