record basic_pair: Type ≝
{ concr: REL;
form: REL;
- rel: arrows ? concr form
+ rel: arrows1 ? concr form
}.
notation "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y}.
interpretation "basic pair relation (non applied)" 'Vdash = (rel _).
record relation_pair (BP1,BP2: basic_pair): Type ≝
- { concr_rel: arrows ? (concr BP1) (concr BP2);
- form_rel: arrows ? (form BP1) (form BP2);
- commute: concr_rel ∘ ⊩ = ⊩ ∘ form_rel
+ { concr_rel: arrows1 ? (concr BP1) (concr BP2);
+ form_rel: arrows1 ? (form BP1) (form BP2);
+ commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
}.
notation "hvbox (r \sub \c)" with precedence 90 for @{'concr_rel $r}.
interpretation "formal relation" 'form_rel r = (form_rel __ r).
definition relation_pair_equality:
- ∀o1,o2. equivalence_relation (relation_pair o1 o2).
+ ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
intros;
constructor 1;
- [ apply (λr,r'. r \sub\c ∘ ⊩ = r' \sub\c ∘ ⊩);
+ [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
| simplify;
intros;
- apply refl;
+ apply refl1;
| simplify;
intros 2;
- apply sym;
+ apply sym1;
| simplify;
intros 3;
- apply trans;
+ apply trans1;
]
qed.
-definition relation_pair_setoid: basic_pair → basic_pair → setoid.
+definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
intros;
constructor 1;
[ apply (relation_pair b b1)
]
qed.
-lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → ⊩ \circ r \sub\f = ⊩ \circ r'\sub\f.
+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;
+ 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);
]
qed.
-definition id: ∀o:basic_pair. relation_pair o o.
+definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
intro;
constructor 1;
- [1,2: apply id;
- | lapply (id_neutral_left ? (concr o) ? (⊩)) as H;
- lapply (id_neutral_right ?? (form o) (⊩)) as H1;
+ [1,2: apply id1;
+ | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
+ lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
apply (.= H);
apply (H1 \sup -1);]
qed.
definition relation_pair_composition:
- ∀o1,o2,o3. binary_morphism (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
+ ∀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);
constructor 1;
- [ apply (r \sub\c ∘ r1 \sub\c)
- | apply (r \sub\f ∘ r1 \sub\f)
+ [ apply (r1 \sub\c ∘ r \sub\c)
+ | apply (r1 \sub\f ∘ r \sub\f)
| lapply (commute ?? r) as H;
lapply (commute ?? r1) as H1;
- apply (.= ASSOC);
- apply (.= #H1);
- apply (.= ASSOC\sup -1);
- apply (.= H#);
- apply comp_assoc]
+ apply (.= ASSOC1);
+ apply (.= #‡H1);
+ apply (.= ASSOC1\sup -1);
+ apply (.= H‡#);
+ apply ASSOC1]
| intros;
- change with (a\sub\c ∘ b\sub\c ∘ ⊩ = a'\sub\c ∘ b'\sub\c ∘ ⊩);
- change in H with (a \sub\c ∘ ⊩ = a' \sub\c ∘ ⊩);
- change in H1 with (b \sub\c ∘ ⊩ = b' \sub\c ∘ ⊩);
- apply (.= ASSOC);
- apply (.= #H1);
- apply (.= #(commute ?? b'));
- apply (.= ASSOC \sup -1);
- apply (.= H#);
- apply (.= ASSOC);
- apply (.= #(commute ?? b')\sup -1);
- apply (ASSOC \sup -1)]
+ change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
+ change in H with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
+ change in H1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
+ apply (.= ASSOC1);
+ apply (.= #‡H1);
+ apply (.= #‡(commute ?? b'));
+ apply (.= ASSOC1 \sup -1);
+ apply (.= H‡#);
+ apply (.= ASSOC1);
+ apply (.= #‡(commute ?? b')\sup -1);
+ apply (ASSOC1 \sup -1)]
qed.
-definition BP: category.
+definition BP: category1.
constructor 1;
[ apply basic_pair
| apply relation_pair_setoid
- | apply id
+ | apply id_relation_pair
| apply relation_pair_composition
| intros;
- change with (a12\sub\c ∘ a23\sub\c ∘ a34\sub\c ∘ ⊩ =
- (a12\sub\c ∘ (a23\sub\c ∘ a34\sub\c) ∘ ⊩));
- apply (ASSOC#);
+ change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
+ ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
+ apply (ASSOC1‡#);
| intros;
- change with ((id o1)\sub\c ∘ a\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
- apply ((id_neutral_left ????)#);
+ change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
+ apply ((id_neutral_right1 ????)‡#);
| intros;
- change with (a\sub\c ∘ (id o2)\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
- apply ((id_neutral_right ????)#);
- ]
-qed.
\ No newline at end of file
+ change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
+ apply ((id_neutral_left1 ????)‡#);]
+qed.
+
+definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
+ intros; constructor 1;
+ [ apply (ext ? ? (rel o));
+ | intros;
+ apply (.= #‡H);
+ apply refl1]
+qed.
+
+definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
+ λo.extS ?? (rel o).
+
+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; split; simplify; intros;
+ [ apply (. #‡((†H)‡(†H1))); assumption
+ | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+qed.
+
+interpretation "fintersects" 'fintersects U V = (fun1 ___ (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; split; simplify; intros;
+ [ apply (. #‡((†H)‡(†H1))); assumption
+ | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+qed.
+
+interpretation "fintersectsS" 'fintersects U V = (fun1 ___ (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 _)).
projects / helm.git / blobdiff