definition composition:
∀A,B,C.
- binary_morphism1 (binary_relation_setoid A B) (binary_relation_setoid B C) (binary_relation_setoid A C).
+ (binary_relation_setoid A B) × (binary_relation_setoid B C) ⇒_1 (binary_relation_setoid A C).
intros;
constructor 1;
[ intros (R12 R23);
∀P,Q. arrows1 REL P Q → binary_relation_setoid P Q ≝ λP,Q,x.x.
coercion binary_relation_setoid_of_arrow1_REL.
-definition full_subset: ∀s:REL. Ω \sup s.
+
+notation > "B ⇒_\r1 C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
+notation "B ⇒\sub (\r 1) C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
+interpretation "'arrows1_SET" 'arrows1_REL A B = (arrows1 REL A B).
+
+
+definition full_subset: ∀s:REL. Ω^s.
apply (λs.{x | True});
intros; simplify; split; intro; assumption.
qed.
coercion full_subset.
-definition comprehension: ∀b:REL. (unary_morphism1 b CPROP) → Ω \sup b.
- apply (λb:REL. λP: b ⇒ CPROP. {x | P x});
+definition comprehension: ∀b:REL. (b ⇒_1. CPROP) → Ω^b.
+ apply (λb:REL. λP: b ⇒_1 CPROP. {x | P x});
intros; simplify;
- alias symbol "trans" = "trans1".
- alias symbol "prop1" = "prop11".
apply (.= †e); apply refl1.
qed.
interpretation "subset comprehension" 'comprehension s p =
(comprehension s (mk_unary_morphism1 ?? p ?)).
-definition ext: ∀X,S:REL. binary_morphism1 (arrows1 ? X S) S (Ω \sup X).
- apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 REL X S.λf:S.{x ∈ X | x ♮r f}) ?);
- [ intros; simplify; apply (.= (e‡#)); apply refl1
+definition ext: ∀X,S:REL. (X ⇒_\r1 S) × S ⇒_1 (Ω^X).
+ intros (X S); constructor 1;
+ [ apply (λr:X ⇒_\r1 S.λf:S.{x ∈ X | x ♮r f}); intros; simplify; apply (.= (e‡#)); apply refl1
| intros; simplify; split; intros; simplify;
[ change with (∀x. x ♮a b → x ♮a' b'); intros;
apply (. (#‡e1^-1)); whd in e; apply (if ?? (e ??)); assumption
| change with (∀x. x ♮a' b' → x ♮a b); intros;
apply (. (#‡e1)); whd in e; apply (fi ?? (e ??));assumption]]
qed.
+
(*
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});*)
*)
(* the same as ⋄ for a basic pair *)
-definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
+definition image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
+ [ apply (λr:U ⇒_\r1 V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
intros; simplify; split; intro; cases e1; exists [1,3: apply w]
[ apply (. (#‡e^-1)‡#); assumption
| apply (. (#‡e)‡#); assumption]
qed.
(* the same as □ for a basic pair *)
-definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
+definition minus_star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
+ [ apply (λr:U ⇒_\r1 V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
intros; simplify; split; intros; apply f;
[ apply (. #‡e); assumption
| apply (. #‡e ^ -1); assumption]
qed.
(* the same as Rest for a basic pair *)
-definition star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+definition star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
+ [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
intros; simplify; split; intros; apply f;
[ apply (. e ‡#); assumption
| apply (. e^ -1‡#); assumption]
qed.
(* the same as Ext for a basic pair *)
-definition minus_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
+ [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
exT ? (λy:V.x ♮r y ∧ y ∈ S) });
intros; simplify; split; intro; cases e1; exists [1,3: apply w]
[ apply (. (e ^ -1‡#)‡#); assumption
apply (if ?? (e ^ -1 ??)); assumption]]
qed.
-(*
(* minus_image is the same as ext *)
theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
intros; unfold image; simplify; split; simplify; intros;
[ change with (a ∈ U);
- cases H; cases x; change in f with (eq1 ? w a); apply (. f‡#); assumption
+ cases e; cases x; change in f with (eq1 ? w a); apply (. f^-1‡#); assumption
| change in f with (a ∈ U);
- exists; [apply a] split; [ change with (a = a); apply refl | assumption]]
+ exists; [apply a] split; [ change with (a = a); apply refl1 | assumption]]
qed.
theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
intros; unfold minus_star_image; simplify; split; simplify; intros;
- [ change with (a ∈ U); apply H; change with (a=a); apply refl
- | change in f1 with (eq1 ? x a); apply (. f1 \sup -1‡#); apply f]
+ [ change with (a ∈ U); apply f; change with (a=a); apply refl1
+ | change in f1 with (eq1 ? x a); apply (. f1‡#); apply f]
qed.
+alias symbol "compose" (instance 2) = "category1 composition".
theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
- intros; unfold image; simplify; split; simplify; intros; cases H; clear H; cases x;
+ intros; unfold image; simplify; split; simplify; intros; cases e; clear e; cases x;
clear x; [ cases f; clear f; | cases f1; clear f1 ]
exists; try assumption; cases x; clear x; split; try assumption;
exists; try assumption; split; assumption.
∀A,B,C,r,s,X.
minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
- [ apply H; exists; try assumption; split; assumption
- | change with (x ∈ X); cases f; cases x1; apply H; assumption]
+ [ apply f; exists; try assumption; split; assumption
+ | change with (x ∈ X); cases f1; cases x1; apply f; assumption]
qed.
+(*
(*CSC: unused! *)
theorem ext_comp:
∀o1,o2,o3: REL.
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