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)
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 e2 (w H3); clear e2; exists; [1,3: apply w ]
- [ apply (. (e‡#)‡(#‡e1)); assumption
- | apply (. ((e \sup -1)‡#)‡(#‡(e1 \sup -1))); assumption]]
+ [ 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) ]
|6,7: intros 5; unfold composition; simplify; split; intro;
unfold setoid1_of_setoid in x y; simplify in x y;
[1,3: cases e (w H1); clear e; cases H1; clear H1; unfold;
- [ apply (. (e ^ -1 : eq1 ? w x)‡#); assumption
- | apply (. #‡(e : eq1 ? w y)); assumption]
+ [ 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.
qed.
coercion full_subset.
-*)
-
-definition setoid1_of_REL: REL → setoid ≝ λS. S.
-coercion setoid1_of_REL.
-
-lemma Type_OF_setoid1_of_REL: ∀o1:Type_OF_category1 REL. Type_OF_objs1 o1 → Type_OF_setoid1 ?(*(setoid1_of_SET o1)*).
- [ apply (setoid1_of_SET o1);
- | intros; apply t;]
-qed.
-coercion Type_OF_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;
(* 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:carr U. x ♮r y ∧ x ∈ S });
+ [ 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)‡#); 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).
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:carr U. x ♮r y → x ∈ S});
+ [ apply (λr: arrows1 ? U 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 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:carr V. x ♮r y → y ∈ S});
+ [ 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 ^ -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.
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:carr V.x ♮r y ∧ y ∈ 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.
| 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 → arrows2 OA (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.
-
-lemma orelation_of_relation_preserves_equality:
- ∀o1,o2:REL.∀t,t': arrows1 ? o1 o2. eq1 ? t t' → orelation_of_relation ?? t = orelation_of_relation ?? t'.
- intros; split; unfold orelation_of_relation; simplify; intro; split; intro;
- simplify; whd in o1 o2;
- [ change with (a1 ∈ minus_star_image ?? t a → a1 ∈ minus_star_image ?? t' a);
- apply (. #‡(e‡#));
- | change with (a1 ∈ minus_star_image ?? t' a → a1 ∈ minus_star_image ?? t a);
- apply (. #‡(e ^ -1‡#));
- | change with (a1 ∈ minus_image ?? t a → a1 ∈ minus_image ?? t' a);
- apply (. #‡(e‡#));
- | change with (a1 ∈ minus_image ?? t' a → a1 ∈ minus_image ?? t a);
- apply (. #‡(e ^ -1‡#));
- | change with (a1 ∈ image ?? t a → a1 ∈ image ?? t' a);
- apply (. #‡(e‡#));
- | change with (a1 ∈ image ?? t' a → a1 ∈ image ?? t a);
- apply (. #‡(e ^ -1‡#));
- | change with (a1 ∈ star_image ?? t a → a1 ∈ star_image ?? t' a);
- apply (. #‡(e‡#));
- | change with (a1 ∈ star_image ?? t' a → a1 ∈ star_image ?? t a);
- apply (. #‡(e ^ -1‡#)); ]
-qed.
-
-lemma hint: ∀o1,o2:OA. Type_OF_setoid2 (arrows2 ? o1 o2) → carr2 (arrows2 OA o1 o2).
- intros; apply t;
-qed.
-coercion hint.
-
-lemma orelation_of_relation_preserves_identity:
- ∀o1:REL. orelation_of_relation ?? (id1 ? o1) = id2 OA (SUBSETS o1).
- intros; split; intro; split; whd; intro;
- [ change with ((∀x. x ♮(id1 REL o1) a1→x∈a) → a1 ∈ a); intros;
- apply (f a1); change with (a1 = a1); apply refl1;
- | change with (a1 ∈ a → ∀x. x ♮(id1 REL o1) a1→x∈a); intros;
- change in f1 with (x = a1); apply (. f1 ^ -1‡#); apply f;
- | alias symbol "and" = "and_morphism".
- change with ((∃y: carr o1.a1 ♮(id1 REL o1) y ∧ y∈a) → a1 ∈ a);
- intro; cases e; clear e; cases x; clear x; change in f with (a1=w);
- apply (. f^-1‡#); apply f1;
- | change with (a1 ∈ a → ∃y: carr o1.a1 ♮(id1 REL o1) y ∧ y∈a);
- intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
- | change with ((∃x: carr o1.x ♮(id1 REL o1) a1∧x∈a) → a1 ∈ a);
- intro; cases e; clear e; cases x; clear x; change in f with (w=a1);
- apply (. f‡#); apply f1;
- | change with (a1 ∈ a → ∃x: carr o1.x ♮(id1 REL o1) a1∧x∈a);
- intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
- | change with ((∀y.a1 ♮(id1 REL o1) y→y∈a) → a1 ∈ a); intros;
- apply (f a1); change with (a1 = a1); apply refl1;
- | change with (a1 ∈ a → ∀y.a1 ♮(id1 REL o1) y→y∈a); intros;
- change in f1 with (a1 = y); apply (. f1‡#); apply f;]
-qed.
-
-lemma hint2: ∀S,T. carr2 (arrows2 OA S T) → Type_OF_setoid2 (arrows2 OA S T).
- intros; apply c;
-qed.
-coercion hint2.
-
-(* CSC: ???? forse un uncertain mancato *)
-lemma orelation_of_relation_preserves_composition:
- ∀o1,o2,o3:REL.∀F: arrows1 ? o1 o2.∀G: arrows1 ? o2 o3.
- orelation_of_relation ?? (G ∘ F) =
- comp2 OA (SUBSETS o1) (SUBSETS o2) (SUBSETS o3)
- ?? (*(orelation_of_relation ?? F) (orelation_of_relation ?? G)*).
- [ apply (orelation_of_relation ?? F); | apply (orelation_of_relation ?? G); ]
- intros; split; intro; split; whd; intro; whd in ⊢ (% → %); intros;
- [ whd; intros; apply f; exists; [ apply x] split; assumption;
- | cases f1; clear f1; cases x1; clear x1; apply (f w); assumption;
- | cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
- split; [ assumption | exists; [apply w] split; assumption ]
- | cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
- split; [ exists; [apply w] split; assumption | assumption ]
- | cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
- split; [ assumption | exists; [apply w] split; assumption ]
- | cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
- split; [ exists; [apply w] split; assumption | assumption ]
- | whd; intros; apply f; exists; [ apply y] split; assumption;
- | cases f1; clear f1; cases x; clear x; apply (f w); assumption;]
-qed.
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