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).
+ (binary_relation_setoid A B) × (binary_relation_setoid B C) ⇒_1 (binary_relation_setoid A C).
intros;
constructor 1;
[ 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) ]
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]
+ [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 full_subset: ∀s:REL. Ω \sup s.
+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.
+
+
+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 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. (b ⇒_1. CPROP) → Ω^b.
+ apply (λb:REL. λP: b ⇒_1 CPROP. {x | P x});
+ intros; simplify;
+ 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]]
+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});*)
intros (X S r); constructor 1;
assumption]
qed.
*)
-(* senza questo exT "fresco", universe inconsistency *)
-inductive exT (A:Type0) (P:A→CProp0) : CProp0 ≝
- ex_introT: ∀w:A. P w → exT A P.
-
-lemma hint: ∀U. carr U → Type_OF_setoid1 ?(*(setoid1_of_SET U)*).
- [ apply setoid1_of_SET; apply U
- | intros; apply c;]
-qed.
-coercion hint.
(* 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*)
- exT ? (λx:carr 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)‡#); assumption
- | apply (. (#‡e ^ -1)‡#); assumption]
+ [ apply (. (#‡e^-1)‡#); assumption
+ | apply (. (#‡e)‡#); assumption]
| intros; split; simplify; intros; cases e2; exists [1,3: apply w]
- [ apply (. #‡(#‡e1)); cases x; split; try assumption;
+ [ apply (. #‡(#‡e1^-1)); cases x; split; try assumption;
apply (if ?? (e ??)); assumption
- | apply (. #‡(#‡e1 ^ -1)); cases x; split; try 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).
+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:carr 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 ^ -1); assumption
- | apply (. #‡e); assumption]
- | intros; split; simplify; intros; [ apply (. #‡e1); | apply (. #‡e1 ^ -1)]
+ [ 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 * for a basic pair *)
-definition star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+(* the same as Rest for a basic pair *)
+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:carr 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 ^ -1‡#); assumption
- | apply (. e‡#); assumption]
- | intros; split; simplify; intros; [ apply (. #‡e1); | apply (. #‡e1 ^ -1)]
+ [ 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 - for a basic pair *)
-definition minus_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+(* the same as Ext for a basic pair *)
+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*)
- exT ? (λy:carr V.x ♮r y ∧ y ∈ 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‡#)‡#); assumption
- | apply (. (e ^ -1‡#)‡#); assumption]
+ [ apply (. (e ^ -1‡#)‡#); assumption
+ | apply (. (e‡#)‡#); assumption]
| intros; split; simplify; intros; cases e2; exists [1,3: apply w]
- [ apply (. #‡(#‡e1)); cases x; split; try assumption;
+ [ apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
apply (if ?? (e ??)); assumption
- | apply (. #‡(#‡e1 ^ -1)); cases x; split; try 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.
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.
| exists; try assumption; split; try assumption; change with (x = x); apply refl]
qed.
*)
-
-include "o-algebra.ma".
-
-definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → ORelation (SUBSETS o1) (SUBSETS o2).
- intros;
- constructor 1;
- [ constructor 1;
- [ apply (λU.image ?? t U);
- | intros; apply (#‡e); ]
- | constructor 1;
- [ apply (λU.minus_star_image ?? t U);
- | intros; apply (#‡e); ]
- | constructor 1;
- [ apply (λU.star_image ?? t U);
- | intros; apply (#‡e); ]
- | constructor 1;
- [ apply (λU.minus_image ?? t U);
- | intros; apply (#‡e); ]
- | intros; split; intro;
- [ change in f with (∀a. a ∈ image ?? t p → a ∈ q);
- change with (∀a:o1. a ∈ p → a ∈ star_image ?? t q);
- intros 4; apply f; exists; [apply a] split; assumption;
- | change in f with (∀a:o1. a ∈ p → a ∈ star_image ?? t q);
- change with (∀a. a ∈ image ?? t p → a ∈ q);
- intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
- | intros; split; intro;
- [ change in f with (∀a. a ∈ minus_image ?? t p → a ∈ q);
- change with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? t q);
- intros 4; apply f; exists; [apply a] split; assumption;
- | change in f with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? t q);
- change with (∀a. a ∈ minus_image ?? t p → a ∈ q);
- intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
- | intros; split; intro; cases f; clear f;
- [ cases x; cases x2; clear x x2; exists; [apply w1]
- [ assumption;
- | exists; [apply w] split; assumption]
- | cases x1; cases x2; clear x1 x2; exists; [apply w1]
- [ exists; [apply w] split; assumption;
- | assumption; ]]]
-qed. sistemare anche l'hint da un'altra parte e capire l'exT (doppio!)
\ No newline at end of file
projects / helm.git / blobdiff