(**************************************************************************)
include "relations.ma".
+include "notation.ma".
record basic_pair: Type1 ≝
{ concr: REL;
rel: arrows1 ? concr form
}.
-notation "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y}.
-notation "⊩" with precedence 60 for @{'Vdash}.
-
-interpretation "basic pair relation" 'Vdash2 x y = (rel _ x y).
-interpretation "basic pair relation (non applied)" 'Vdash = (rel _).
+interpretation "basic pair relation" 'Vdash2 x y c = (fun21 ___ (rel c) x y).
+interpretation "basic pair relation (non applied)" 'Vdash c = (rel c).
alias symbol "eq" = "setoid1 eq".
alias symbol "compose" = "category1 composition".
commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
}.
-notation "hvbox (r \sub \c)" with precedence 90 for @{'concr_rel $r}.
-notation "hvbox (r \sub \f)" with precedence 90 for @{'form_rel $r}.
interpretation "concrete relation" 'concr_rel r = (concr_rel __ r).
interpretation "formal relation" 'form_rel r = (form_rel __ r).
]
qed.
-lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
+definition relation_pair_of_relation_pair_setoid :
+ ∀P,Q. relation_pair_setoid P Q → relation_pair P Q ≝ λP,Q,x.x.
+coercion relation_pair_of_relation_pair_setoid.
+
+lemma eq_to_eq':
+ ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
intros 7 (o1 o2 r r' H c1 f2);
split; intro H1;
[ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
apply (H1 \sup -1);]
qed.
-definition relation_pair_composition:
- ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
- intros;
- constructor 1;
- [ intros (r r1);
+lemma relation_pair_composition:
+ ∀o1,o2,o3: basic_pair.
+ relation_pair_setoid o1 o2 → relation_pair_setoid o2 o3 → relation_pair_setoid o1 o3.
+intros 3 (o1 o2 o3);
+ intros (r r1);
constructor 1;
[ apply (r1 \sub\c ∘ r \sub\c)
| apply (r1 \sub\f ∘ r \sub\f)
apply (.= ASSOC ^ -1);
apply (.= H‡#);
apply ASSOC]
- | intros;
+qed.
+
+lemma relation_pair_composition_is_morphism:
+ ∀o1,o2,o3: basic_pair.
+ ∀a,a':relation_pair_setoid o1 o2.
+ ∀b,b':relation_pair_setoid o2 o3.
+ a=a' → b=b' →
+ relation_pair_composition o1 o2 o3 a b
+ = relation_pair_composition o1 o2 o3 a' b'.
+intros 3 (o1 o2 o3);
+ intros;
change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
apply (.= e‡#);
apply (.= ASSOC);
apply (.= #‡(commute ?? b')\sup -1);
- apply (ASSOC ^ -1)]
+ apply (ASSOC ^ -1);
qed.
-
-definition BP: category1.
+
+definition relation_pair_composition_morphism:
+ ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
+ intros;
constructor 1;
- [ apply basic_pair
- | apply relation_pair_setoid
- | apply id_relation_pair
- | apply relation_pair_composition
- | intros;
+ [ apply relation_pair_composition;
+ | apply relation_pair_composition_is_morphism;]
+qed.
+
+lemma relation_pair_composition_morphism_assoc:
+Πo1:basic_pair
+.Πo2:basic_pair
+ .Πo3:basic_pair
+ .Πo4:basic_pair
+ .Πa12:relation_pair_setoid o1 o2
+ .Πa23:relation_pair_setoid o2 o3
+ .Πa34:relation_pair_setoid o3 o4
+ .relation_pair_composition_morphism o1 o3 o4
+ (relation_pair_composition_morphism o1 o2 o3 a12 a23) a34
+ =relation_pair_composition_morphism o1 o2 o4 a12
+ (relation_pair_composition_morphism o2 o3 o4 a23 a34).
+ intros;
change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
alias symbol "refl" = "refl1".
alias symbol "prop2" = "prop21".
apply (ASSOC‡#);
- | intros;
+qed.
+
+lemma relation_pair_composition_morphism_respects_id:
+ ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
+ relation_pair_composition_morphism o1 o1 o2 (id_relation_pair o1) a=a.
+ intros;
change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
- apply ((id_neutral_right1 ????)‡#);
- | intros;
+ apply ((id_neutral_right1 ????)‡#);
+qed.
+
+lemma relation_pair_composition_morphism_respects_id_r:
+ ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
+ relation_pair_composition_morphism o1 o2 o2 a (id_relation_pair o2)=a.
+ intros;
change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
- apply ((id_neutral_left1 ????)‡#);]
+ apply ((id_neutral_left1 ????)‡#);
+qed.
+
+definition BP: category1.
+ constructor 1;
+ [ apply basic_pair
+ | apply relation_pair_setoid
+ | apply id_relation_pair
+ | apply relation_pair_composition_morphism
+ | apply relation_pair_composition_morphism_assoc;
+ | apply relation_pair_composition_morphism_respects_id;
+ | apply relation_pair_composition_morphism_respects_id_r;]
qed.
+
+definition basic_pair_of_BP : objs1 BP → basic_pair ≝ λx.x.
+coercion basic_pair_of_BP.
+
+definition relation_pair_setoid_of_arrows1_BP :
+ ∀P,Q. arrows1 BP P Q → relation_pair_setoid P Q ≝ λP,Q,x.x.
+coercion relation_pair_setoid_of_arrows1_BP.
-(*
definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
intros; constructor 1;
[ apply (ext ? ? (rel o));
| intros;
- apply (.= #‡H);
+ apply (.= #‡e);
apply refl1]
qed.
-definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
- λo.extS ?? (rel o).
+definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o).
+ intros; constructor 1;
+ [ apply (minus_image ?? (rel o));
+ | intros; apply (#‡e); ]
+qed.
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; simplify; apply (.= (†e)‡#); apply refl1
| intros; split; simplify; intros;
- [ apply (. #‡((†H)‡(†H1))); assumption
- | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+ [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
+ | apply (. #‡((†e)‡(†e1))); assumption]]
qed.
-interpretation "fintersects" 'fintersects U V = (fun1 ___ (fintersects _) U V).
+interpretation "fintersects" 'fintersects U V = (fun21 ___ (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; simplify; apply (.= (†e)‡#); apply refl1
| intros; split; simplify; intros;
- [ apply (. #‡((†H)‡(†H1))); assumption
- | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+ [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
+ | apply (. #‡((†e)‡(†e1))); assumption]]
qed.
-interpretation "fintersectsS" 'fintersects U V = (fun1 ___ (fintersectsS _) U V).
+interpretation "fintersectsS" 'fintersects U V = (fun21 ___ (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 _)).
-*)
-
-include "o-basic_pairs.ma".
-(* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
-definition o_basic_pair_of_basic_pair: cic:/matita/formal_topology/basic_pairs/basic_pair.ind#xpointer(1/1) → basic_pair.
- intro;
- constructor 1;
- [ apply (SUBSETS (concr b));
- | apply (SUBSETS (form b));
- | apply (orelation_of_relation ?? (rel b)); ]
+ [ apply (λx:concr o.λS: Ω \sup (form o).∃y:form o.y ∈ S ∧ x ⊩_o y);
+ | intros; split; intros; cases e2; exists [1,3: apply w]
+ [ apply (. (#‡e1^-1)‡(e^-1‡#)); assumption
+ | apply (. (#‡e1)‡(e‡#)); assumption]]
qed.
-definition o_relation_pair_of_relation_pair:
- ∀BP1,BP2.cic:/matita/formal_topology/basic_pairs/relation_pair.ind#xpointer(1/1) BP1 BP2 →
- relation_pair (o_basic_pair_of_basic_pair BP1) (o_basic_pair_of_basic_pair BP2).
- intros;
- constructor 1;
- [ apply (orelation_of_relation ?? (r \sub \c));
- | apply (orelation_of_relation ?? (r \sub \f));
- | lapply (commute ?? r);
- lapply (orelation_of_relation_preserves_equality ???? Hletin);
- apply (.= (orelation_of_relation_preserves_composition (concr BP1) ??? (rel BP2)) ^ -1);
- apply (.= (orelation_of_relation_preserves_equality ???? (commute ?? r)));
- apply (orelation_of_relation_preserves_composition ?? (form BP2) (rel BP1) ?); ]
-qed.
\ No newline at end of file
+interpretation "basic pair relation for subsets" 'Vdash2 x y c = (fun21 (concr _) __ (relS c) x y).
+interpretation "basic pair relation for subsets (non applied)" 'Vdash c = (fun21 ___ (relS c)).
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