interpretation "subseteq" 'subseteq U V = (fun1 ___ (subseteq _) U V).
+theorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
+ intros 4; assumption.
+qed.
+
+theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
+ intros; apply transitive_subseteq_operator; [apply S2] assumption.
+qed.
+
definition overlaps: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
intros;
constructor 1;
interpretation "union" 'union U V = (fun1 ___ (union _) U V).
-definition singleton: ∀A:setoid. A → Ω \sup A.
- apply (λA:setoid.λa:A.{b | a=b});
- intros; simplify;
- split; intro;
- apply (.= H1);
- [ apply H | apply (H \sup -1) ]
+definition singleton: ∀A:setoid. unary_morphism A (Ω \sup A).
+ intros; constructor 1;
+ [ apply (λA:setoid.λa:A.{b | a=b});
+ intros; simplify;
+ split; intro;
+ apply (.= H1);
+ [ apply H | apply (H \sup -1) ]
+ | intros; split; intros 2; simplify in f ⊢ %; apply trans;
+ [ apply a |4: apply a'] try assumption; apply sym; assumption]
qed.
-interpretation "singleton" 'singl a = (singleton _ a).
+interpretation "singleton" 'singl a = (fun_1 __ (singleton _) a).