notation < "hvbox (x \nbsp \natur term 90 r \nbsp y)" with precedence 45 for @{'satisfy $r $x $y}.
notation > "hvbox (x \natur term 90 r y)" with precedence 45 for @{'satisfy $r $x $y}.
-interpretation "relation applied" 'satisfy r x y = (fun21 ___ (satisfy __ r) x y).
+interpretation "relation applied" 'satisfy r x y = (fun21 ??? (satisfy ?? r) x y).
-definition binary_relation_setoid: SET → SET → SET1.
+definition binary_relation_setoid: SET → SET → setoid1.
intros (A B);
constructor 1;
[ apply (binary_relation A B)
[ apply (λA,B.λr,r': binary_relation A B. ∀x,y. r x y ↔ r' x y)
| simplify; intros 3; split; intro; assumption
| simplify; intros 5; split; intro;
- [ apply (fi ?? (H ??)) | apply (if ?? (H ??))] assumption
+ [ apply (fi ?? (f ??)) | apply (if ?? (f ??))] assumption
| simplify; intros 7; split; intro;
- [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
- [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
+ [ apply (if ?? (f1 ??)) | apply (fi ?? (f ??)) ]
+ [ apply (if ?? (f ??)) | apply (fi ?? (f1 ??)) ]
assumption]]
qed.
+definition binary_relation_of_binary_relation_setoid :
+ ∀A,B.binary_relation_setoid A B → binary_relation A B ≝ λA,B,c.c.
+coercion binary_relation_of_binary_relation_setoid.
+
definition composition:
∀A,B,C.
binary_morphism1 (binary_relation_setoid A B) (binary_relation_setoid B C) (binary_relation_setoid A C).
[ intros (R12 R23);
constructor 1;
constructor 1;
- [ alias symbol "and" = "and_morphism".
- (* carr to avoid universe inconsistency *)
- apply (λs1:carr A.λs3:carr C.∃s2:carr B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
+ [ apply (λs1:A.λs3:C.∃s2:B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
| intros;
- split; intro; cases H (w H3); clear H; exists; [1,3: apply w ]
- [ apply (. (e‡#)‡(#‡e1)); assumption
- | apply (. ((e \sup -1)‡#)‡(#‡(e1 \sup -1))); assumption]]
+ split; intro; cases e2 (w H3); clear e2; exists; [1,3: apply w ]
+ [ apply (. (e^-1‡#)‡(#‡e1^-1)); assumption
+ | apply (. (e‡#)‡(#‡e1)); assumption]]
| intros 8; split; intro H2; simplify in H2 ⊢ %;
cases H2 (w H3); clear H2; exists [1,3: apply w] cases H3 (H2 H4); clear H3;
[ lapply (if ?? (e x w) H2) | lapply (fi ?? (e x w) H2) ]
exists; try assumption;
split; assumption]
qed.
-axiom daemon: False.
+
definition REL: category1.
constructor 1;
[ apply setoid
| intros (T T1); apply (binary_relation_setoid T T1)
| intros; constructor 1;
constructor 1; unfold setoid1_of_setoid; simplify;
- [ change with (carr o → carr o → CProp); intros; apply (eq1 ? c c1) ]] cases daemon; qed.
- | intros; split; intro;
+ [ (* changes required to avoid universe inconsistency *)
+ change with (carr o → carr o → CProp); intros; apply (eq ? c c1)
+ | intros; split; intro; change in a a' b b' with (carr o);
+ change in e with (eq ? a a'); change in e1 with (eq ? b b');
[ apply (.= (e ^ -1));
apply (.= e2);
apply e1
split; assumption
|6,7: intros 5; unfold composition; simplify; split; intro;
unfold setoid1_of_setoid in x y; simplify in x y;
- [1,3: cases H (w H1); clear H; cases H1; clear H1; unfold;
- [ apply (. (e ^ -1 : eq1 ? w x)‡#); assumption
- | apply (. #‡(e : eq1 ? w y)); assumption]
- |2,4: exists; try assumption; split; first [apply refl1 | assumption]]]
+ [1,3: cases e (w H1); clear e; cases H1; clear H1; unfold;
+ [ apply (. (e : eq1 ? x w)‡#); assumption
+ | apply (. #‡(e : eq1 ? w y)^-1); assumption]
+ |2,4: exists; try assumption; split;
+ (* change required to avoid universe inconsistency *)
+ change in x with (carr o1); change in y with (carr o2);
+ first [apply refl | assumption]]]
qed.
+definition setoid_of_REL : objs1 REL → setoid ≝ λx.x.
+coercion setoid_of_REL.
+
+definition binary_relation_setoid_of_arrow1_REL :
+ ∀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.
apply (λs.{x | True});
intros; simplify; split; intro; assumption.
coercion full_subset.
-definition setoid1_of_REL: REL → setoid ≝ λS. S.
-
-coercion setoid1_of_REL.
-
-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.
+definition comprehension: ∀b:REL. (unary_morphism1 b CPROP) → Ω \sup b.
+ apply (λb:REL. λP: b ⇒ 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_morphism __ 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 ? X S.λf:S.{x ∈ X | x ♮r f}) ?);
- [ intros; simplify; apply (.= (H‡#)); apply refl1
- | intros; simplify; split; intros; simplify; intros; cases f; split; try assumption;
- [ apply (. (#‡H1)); whd in H; apply (if ?? (H ??)); assumption
- | apply (. (#‡H1\sup -1)); whd in H; apply (fi ?? (H ??));assumption]]
+ apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 REL X 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});*)
intros (X S r); constructor 1;
cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
assumption]
qed.
+*)
(* the same as ⋄ for a basic pair *)
definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S});
- intros; simplify; split; intro; cases H1; exists [1,3: apply w]
- [ apply (. (#‡H)‡#); assumption
- | apply (. (#‡H \sup -1)‡#); assumption]
- | intros; split; simplify; intros; cases H2; exists [1,3: apply w]
- [ apply (. #‡(#‡H1)); cases x; split; try assumption;
- apply (if ?? (H ??)); assumption
- | apply (. #‡(#‡H1 \sup -1)); cases x; split; try assumption;
- apply (if ?? (H \sup -1 ??)); assumption]]
+ [ apply (λr: arrows1 ? U 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]
+ | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
+ [ apply (. #‡(#‡e1^-1)); cases x; split; try assumption;
+ apply (if ?? (e ??)); assumption
+ | apply (. #‡(#‡e1)); cases x; split; try assumption;
+ apply (if ?? (e ^ -1 ??)); 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).
intros; constructor 1;
[ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
- intros; simplify; split; intros; apply H1;
- [ apply (. #‡H \sup -1); assumption
- | apply (. #‡H); assumption]
- | intros; split; simplify; intros; [ apply (. #‡H1); | apply (. #‡H1 \sup -1)]
- apply H2; [ apply (if ?? (H \sup -1 ??)); | apply (if ?? (H ??)) ] assumption]
+ intros; simplify; split; intros; apply f;
+ [ apply (. #‡e); assumption
+ | apply (. #‡e ^ -1); assumption]
+ | intros; split; simplify; intros; [ apply (. #‡e1^ -1); | apply (. #‡e1 )]
+ apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] 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).
+ intros; constructor 1;
+ [ apply (λr: arrows1 ? U 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]
+ | intros; split; simplify; intros; [ apply (. #‡e1 ^ -1); | apply (. #‡e1)]
+ apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] 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).
+ intros; constructor 1;
+ [ apply (λr: arrows1 ? U 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 (. (e‡#)‡#); assumption]
+ | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
+ [ apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
+ apply (if ?? (e ??)); assumption
+ | apply (. #‡(#‡e1)); cases x; split; try 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.
[ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
assumption
| exists; try assumption; split; try assumption; change with (x = x); apply refl]
-qed.
\ No newline at end of file
+qed.
+*)
projects / helm.git / blobdiff