-definition comprehension: ∀b:REL. (b ⇒ CPROP) → Ω \sup b.
- apply (λb:REL. λP: b ⇒ CPROP. {x | x ∈ b ∧ P x});
- intros; simplify; apply (.= (H‡#)‡(†H)); apply refl1.
-qed.
-
-interpretation "subset comprehension" 'comprehension s p =
- (comprehension s (mk_unary_morphism __ p _)).
-
-definition ext: ∀X,S:REL. ∀r: arrows1 ? X S. S ⇒ Ω \sup X.
- apply (λX,S,r.mk_unary_morphism ?? (λf.{x ∈ X | x ♮r f}) ?);
- [ intros; simplify; apply (.= (H‡#)); apply refl1
- | intros; simplify; split; intros; simplify; intros;
- [ apply (. #‡(#‡H)); assumption
- | apply (. #‡(#‡H\sup -1)); assumption]]
-qed.
-
-definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o) ≝ λo.ext ? ? (rel o).
-
-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});*)
- intros (X S r); constructor 1;
- [ intro F; constructor 1; constructor 1;
- [ apply (λx. x ∈ X ∧ ∃a:S. a ∈ F ∧ x ♮r a);
- | intros; split; intro; cases f (H1 H2); clear f; split;
- [ apply (. (H‡#)); assumption
- |3: apply (. (H\sup -1‡#)); assumption
- |2,4: cases H2 (w H3); exists; [1,3: apply w]
- [ apply (. (#‡(H‡#))); assumption
- | apply (. (#‡(H \sup -1‡#))); assumption]]]
- | intros; split; simplify; intros; cases f; cases H1; split;
- [1,3: assumption
- |2,4: exists; [1,3: apply w]
- [ apply (. (#‡H)‡#); assumption
- | apply (. (#‡H\sup -1)‡#); assumption]]]
-qed.
-
-definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
- λo.extS ?? (rel o).
-
-definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
- intros (o); constructor 1;
- [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
- intros; simplify; apply (.= (†H)‡#); apply refl1
- | intros; split; simplify; intros;
- [ apply (. #‡((†H)‡(†H1))); assumption
- | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
-qed.
-
-interpretation "fintersects" 'fintersects U V = (fun1 ___ (fintersects _) U V).
-
-definition fintersectsS:
- ∀o:BP. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
- intros (o); constructor 1;
- [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
- intros; simplify; apply (.= (†H)‡#); apply refl1
- | intros; split; simplify; intros;
- [ apply (. #‡((†H)‡(†H1))); assumption
- | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
-qed.
-
-interpretation "fintersectsS" 'fintersects U V = (fun1 ___ (fintersectsS _) U V).
-
-definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
- intros (o); constructor 1;
- [ apply (λx:concr o.λS: Ω \sup (form o).∃y: form o.y ∈ S ∧ x ⊩ y);
- | intros; split; intros; cases H2; exists [1,3: apply w]
- [ apply (. (#‡H1)‡(H‡#)); assumption
- | apply (. (#‡H1 \sup -1)‡(H \sup -1‡#)); assumption]]
-qed.
-
-interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr _) __ (relS _) x y).
-interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ___ (relS _)).
-