[ apply (λA:objs1 SET.λU,V:Ω \sup A.(exT2 ? (λx:A.?(*x*) ∈ U) (λx:A.?(*x*) ∈ V) : CProp0))
| intros;
constructor 1; intro; cases e2; exists; [1,4: apply w]
- [ apply (. #‡e); assumption
- | apply (. #‡e1); assumption
- | apply (. #‡(e \sup -1)); assumption;
- | apply (. #‡(e1 \sup -1)); assumption]]
+ [ apply (. #‡e^-1); assumption
+ | apply (. #‡e1^-1); assumption
+ | apply (. #‡e); assumption;
+ | apply (. #‡e1); assumption]]
qed.
interpretation "overlaps" 'overlaps U V = (fun21 ___ (overlaps _) U V).
intros; simplify; apply (.= (e‡#)‡(e‡#)); apply refl1;
| intros;
split; intros 2; simplify in f ⊢ %;
- [ apply (. (#‡e)‡(#‡e1)); assumption
- | apply (. (#‡(e \sup -1))‡(#‡(e1 \sup -1))); assumption]]
+ [ apply (. (#‡e^-1)‡(#‡e1^-1)); assumption
+ | apply (. (#‡e)‡(#‡e1)); assumption]]
qed.
interpretation "intersects" 'intersects U V = (fun21 ___ (intersects _) U V).
intros; simplify; apply (.= (e‡#)‡(e‡#)); apply refl1
| intros;
split; intros 2; simplify in f ⊢ %;
- [ apply (. (#‡e)‡(#‡e1)); assumption
- | apply (. (#‡(e \sup -1))‡(#‡(e1 \sup -1))); assumption]]
+ [ apply (. (#‡e^-1)‡(#‡e1^-1)); assumption
+ | apply (. (#‡e)‡(#‡e1)); assumption]]
qed.
interpretation "union" 'union U V = (fun21 ___ (union _) U V).
(* qua non riesco a mettere set *)
definition singleton: ∀A:setoid. unary_morphism1 A (Ω \sup A).
intros; constructor 1;
- [ apply (λa:A.{b | eq ? a b}); unfold setoid1_of_setoid; simplify;
+ [ apply (λa:A.{b | a =_0 b}); unfold setoid1_of_setoid; simplify;
intros; simplify;
split; intro;
apply (.= e1);
∀A:SET.∀I:SET.unary_morphism2 (setoid1_of_setoid I ⇒ Ω \sup A) (setoid2_of_setoid1 (Ω \sup A)).
intros; constructor 1;
[ intro; whd; whd in I;
- apply ({x | ∀i:I. x ∈ t i});
- simplify; intros; split; intros; [ apply (. (e‡#)); | apply (. (e \sup -1‡#)); ]
+ apply ({x | ∀i:I. x ∈ c i});
+ simplify; intros; split; intros; [ apply (. (e^-1‡#)); | apply (. e‡#); ]
apply f;
| intros; split; intros 2; simplify in f ⊢ %; intro;
- [ apply (. (#‡(e i))); apply f;
- | apply (. (#‡(e i)\sup -1)); apply f]]
+ [ apply (. (#‡(e i)^-1)); apply f;
+ | apply (. (#‡e i)); apply f]]
qed.
definition big_union:
∀A:SET.∀I:SET.unary_morphism2 (setoid1_of_setoid I ⇒ Ω \sup A) (setoid2_of_setoid1 (Ω \sup A)).
intros; constructor 1;
[ intro; whd; whd in A; whd in I;
- apply ({x | ∃i:carr I. x ∈ t i });
+ apply ({x | ∃i:I. x ∈ c i });
simplify; intros; split; intros; cases e1; clear e1; exists; [1,3:apply w]
- [ apply (. (e‡#)); | apply (. (e \sup -1‡#)); ]
+ [ apply (. (e^-1‡#)); | apply (. (e‡#)); ]
apply x;
| intros; split; intros 2; simplify in f ⊢ %; cases f; clear f; exists; [1,3:apply w]
- [ apply (. (#‡(e w))); apply x;
- | apply (. (#‡(e w)\sup -1)); apply x]]
+ [ apply (. (#‡(e w)^-1)); apply x;
+ | apply (. (#‡e w)); apply x]]
qed.