assumption.
qed.
+(*
+
definition powerset_setoid: setoid → setoid1.
intros (T);
constructor 1;
| apply s1; assumption]]
qed.
-interpretation "mem" 'mem a S = (fun1 ___ (mem _) a S).
+interpretation "mem" 'mem a S = (fun1 ??? (mem ?) a S).
definition subseteq: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
intros;
apply (transitive_subseteq_operator ???? s s4) ]]
qed.
-interpretation "subseteq" 'subseteq U V = (fun1 ___ (subseteq _) U V).
+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;
| apply (. #‡(H1 \sup -1)); assumption]]
qed.
-interpretation "overlaps" 'overlaps U V = (fun1 ___ (overlaps _) U V).
+interpretation "overlaps" 'overlaps U V = (fun1 ??? (overlaps ?) U V).
definition intersects:
∀A. binary_morphism1 (powerset_setoid A) (powerset_setoid A) (powerset_setoid A).
| apply (. (#‡(H \sup -1))‡(#‡(H1 \sup -1))); assumption]]
qed.
-interpretation "intersects" 'intersects U V = (fun1 ___ (intersects _) U V).
+interpretation "intersects" 'intersects U V = (fun1 ??? (intersects ?) U V).
definition union:
∀A. binary_morphism1 (powerset_setoid A) (powerset_setoid A) (powerset_setoid A).
| apply (. (#‡(H \sup -1))‡(#‡(H1 \sup -1))); assumption]]
qed.
-interpretation "union" 'union U V = (fun1 ___ (union _) U V).
+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).
+
+*)
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