include "relations.ma".
include "notation.ma".
-record basic_pair: Type1 ≝
- { concr: REL;
- form: REL;
- rel: arrows1 ? concr form
- }.
+record basic_pair: Type1 ≝ {
+ concr: REL; form: REL; rel: concr ⇒_\r1 form
+}.
-interpretation "basic pair relation" 'Vdash2 x y c = (fun21 ___ (rel c) x y).
+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".
-record relation_pair (BP1,BP2: basic_pair): Type1 ≝
- { concr_rel: arrows1 ? (concr BP1) (concr BP2);
- form_rel: arrows1 ? (form BP1) (form BP2);
- commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
+record relation_pair (BP1,BP2: basic_pair): Type1 ≝ {
+ concr_rel: (concr BP1) ⇒_\r1 (concr BP2); form_rel: (form BP1) ⇒_\r1 (form BP2);
+ commute: ⊩ ∘ concr_rel =_1 form_rel ∘ ⊩
}.
+interpretation "concrete relation" 'concr_rel r = (concr_rel ?? r).
+interpretation "formal relation" 'form_rel r = (form_rel ?? r).
-interpretation "concrete relation" 'concr_rel r = (concr_rel __ r).
-interpretation "formal relation" 'form_rel r = (form_rel __ r).
-
-definition relation_pair_equality:
- ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
- intros;
- constructor 1;
- [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
- | simplify;
- intros;
- apply refl1;
- | simplify;
- intros 2;
- apply sym1;
- | simplify;
- intros 3;
- apply trans1;
- ]
+definition relation_pair_equality: ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
+ intros; constructor 1; [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
+ | simplify; intros; apply refl1;
+ | simplify; intros 2; apply sym1;
+ | simplify; intros 3; apply trans1; ]
qed.
definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
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;
- lapply (if ?? (H c1 f2) H2) as H3;
- apply (if ?? (commute ?? r' c1 f2) H3);
- | lapply (fi ?? (commute ?? r' c1 f2) H1) as H2;
- lapply (fi ?? (H c1 f2) H2) as H3;
- apply (if ?? (commute ?? r c1 f2) H3);
- ]
+ ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r =_1 r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
+ intros 5 (o1 o2 r r' H);
+ apply (.= (commute ?? r)^-1);
+ change in H with (⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
+ apply rule (.= H);
+ apply (commute ?? r').
qed.
definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
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.
∀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).
+(*
+definition BPext: ∀o: BP. (form o) ⇒_1 Ω^(concr o).
intros; constructor 1;
[ apply (ext ? ? (rel o));
| intros;
apply refl1]
qed.
-definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o).
+definition BPextS: ∀o: BP. Ω^(form o) ⇒_1 Ω^(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)).
+definition fintersects: ∀o: BP. (form o) × (form o) ⇒_1 Ω^(form o).
intros (o); constructor 1;
[ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
intros; simplify; apply (.= (†e)‡#); apply refl1
| apply (. #‡((†e)‡(†e1))); assumption]]
qed.
-interpretation "fintersects" 'fintersects U V = (fun21 ___ (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)).
+ ∀o:BP. Ω^(form o) × Ω^(form o) ⇒_1 Ω^(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 });
+ [ apply (λo: basic_pair.λa,b: Ω^(form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
intros; simplify; apply (.= (†e)‡#); apply refl1
| intros; split; simplify; intros;
[ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
| apply (. #‡((†e)‡(†e1))); assumption]]
qed.
-interpretation "fintersectsS" 'fintersects U V = (fun21 ___ (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.
+definition relS: ∀o: BP. (concr o) × Ω^(form o) ⇒_1 CPROP.
intros (o); constructor 1;
- [ apply (λx:concr o.λS: Ω \sup (form o).∃y:form o.y ∈ S ∧ x ⊩_o y);
+ [ apply (λx:concr o.λS: Ω^(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.
-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)).
+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)).
+*)
projects / helm.git / blobdiff