change with (a\sub\c ∘ (id_relation_pair o2)\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
apply ((id_neutral_right1 ????)‡#);
]
-qed.
\ No newline at end of file
+qed.
+
+definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o) ≝ λo.ext ? ? (rel o).
+
+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 _)).
include "formal_topology/basic_pairs.ma".
-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.
-qed.
-
-interpretation "subset comprehension" 'comprehension s p =
- (comprehension s (mk_unary_morphism __ p _)).
-
-definition ext: ∀X,S:REL. ∀r: arrows1 ? X S. S ⇒ Ω \sup X.
- apply (λX,S,r.mk_unary_morphism ?? (λf.{x ∈ X | x ♮r f}) ?);
- [ intros; simplify; apply (.= (H‡#)); apply refl1
- | intros; simplify; split; intros; simplify; intros;
- [ apply (. #‡(#‡H)); assumption
- | apply (. #‡(#‡H\sup -1)); assumption]]
-qed.
-
-definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o) ≝ λo.ext ? ? (rel o).
-
-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;
- [ intro F; constructor 1; constructor 1;
- [ apply (λx. x ∈ X ∧ ∃a:S. a ∈ F ∧ x ♮r a);
- | intros; split; intro; cases f (H1 H2); clear f; split;
- [ apply (. (H‡#)); assumption
- |3: apply (. (H\sup -1‡#)); assumption
- |2,4: cases H2 (w H3); exists; [1,3: apply w]
- [ apply (. (#‡(H‡#))); assumption
- | apply (. (#‡(H \sup -1‡#))); assumption]]]
- | intros; split; simplify; intros; cases f; cases H1; split;
- [1,3: assumption
- |2,4: exists; [1,3: apply w]
- [ apply (. (#‡H)‡#); assumption
- | apply (. (#‡H\sup -1)‡#); assumption]]]
-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 _)).
-
record concrete_space : Type ≝
{ bp:> BP;
converges: ∀a: concr bp.∀U,V: form bp. a ⊩ U → a ⊩ V → a ⊩ (U ↓ V);
coercion rp''.
-lemma equalset_extS_id_X_X: ∀o:REL.∀X.extS ?? (id1 ? o) X = X.
- intros;
- unfold extS; simplify;
- split; simplify;
- [ intros 2; change with (a ∈ X);
- cases f; clear f;
- cases H; clear H;
- cases x; clear x;
- change in f2 with (eq1 ? a w);
- apply (. (f2\sup -1‡#));
- assumption
- | intros 2; change in f with (a ∈ X);
- split;
- [ whd; exact I
- | exists; [ apply a ]
- split;
- [ assumption
- | change with (a = a); apply refl]]]
-qed.
-
-lemma extS_id: ∀o:BP.∀X.extS (concr o) (concr o) (id1 ? o) \sub \c X = X.
- intros;
- unfold extS; simplify;
- split; simplify; intros;
- [ change with (a ∈ X);
- cases f; cases H; cases x; change in f3 with (eq1 ? a w);
- apply (. (f3\sup -1‡#));
- assumption
- | change in f with (a ∈ X);
- split;
- [ apply I
- | exists; [apply a]
- split; [ assumption | change with (a = a); apply refl]]]
-qed.
-
-lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c1 ∘ c2) S = extS o1 o2 c1 (extS o2 o3 c2 S).
- intros; unfold extS; simplify; split; intros; simplify; intros;
- [ cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
- cases H3 (H4 H5); cases H5 (w1 H6); clear H3 H5; cases H6 (H7 H8); clear H6;
- exists; [apply w1] split [2: assumption] constructor 1; [assumption]
- exists; [apply w] split; assumption
- | cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
- cases H3 (H4 H5); cases H4 (w1 H6); clear H3 H4; cases H6 (w2 H7); clear H6;
- cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
- assumption]
-qed.
-
definition convergent_relation_space_composition:
∀o1,o2,o3: concrete_space.
binary_morphism1
apply (.= (†((equalset_extS_id_X_X ??)\sup -1‡
(equalset_extS_id_X_X ??)\sup -1)));
apply refl1;
- | apply (.= (extS_id ??));
+ | apply (.= (equalset_extS_id_X_X ??));
apply refl1]
| apply convergent_relation_space_composition
| intros; simplify;
change with (a ∘ id1 ? o2 = a);
apply (.= id_neutral_right1 ????);
apply refl1]
-qed.
\ No newline at end of file
+qed.
definition setoid1_of_REL: REL → setoid ≝ λS. S.
-coercion setoid1_of_REL.
\ No newline at end of file
+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.
+qed.
+
+interpretation "subset comprehension" 'comprehension s p =
+ (comprehension s (mk_unary_morphism __ p _)).
+
+definition ext: ∀X,S:REL. ∀r: arrows1 ? X S. S ⇒ Ω \sup X.
+ apply (λX,S,r.mk_unary_morphism ?? (λf.{x ∈ X | x ♮r f}) ?);
+ [ intros; simplify; apply (.= (H‡#)); apply refl1
+ | intros; simplify; split; intros; simplify; intros;
+ [ apply (. #‡(#‡H)); assumption
+ | apply (. #‡(#‡H\sup -1)); 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;
+ [ intro F; constructor 1; constructor 1;
+ [ apply (λx. x ∈ X ∧ ∃a:S. a ∈ F ∧ x ♮r a);
+ | intros; split; intro; cases f (H1 H2); clear f; split;
+ [ apply (. (H‡#)); assumption
+ |3: apply (. (H\sup -1‡#)); assumption
+ |2,4: cases H2 (w H3); exists; [1,3: apply w]
+ [ apply (. (#‡(H‡#))); assumption
+ | apply (. (#‡(H \sup -1‡#))); assumption]]]
+ | intros; split; simplify; intros; cases f; cases H1; split;
+ [1,3: assumption
+ |2,4: exists; [1,3: apply w]
+ [ apply (. (#‡H)‡#); assumption
+ | apply (. (#‡H\sup -1)‡#); assumption]]]
+qed.
+
+lemma equalset_extS_id_X_X: ∀o:REL.∀X.extS ?? (id1 ? o) X = X.
+ intros;
+ unfold extS; simplify;
+ split; simplify;
+ [ intros 2; change with (a ∈ X);
+ cases f; clear f;
+ cases H; clear H;
+ cases x; clear x;
+ change in f2 with (eq1 ? a w);
+ apply (. (f2\sup -1‡#));
+ assumption
+ | intros 2; change in f with (a ∈ X);
+ split;
+ [ whd; exact I
+ | exists; [ apply a ]
+ split;
+ [ assumption
+ | change with (a = a); apply refl]]]
+qed.
+
+lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c1 ∘ c2) S = extS o1 o2 c1 (extS o2 o3 c2 S).
+ intros; unfold extS; simplify; split; intros; simplify; intros;
+ [ cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
+ cases H3 (H4 H5); cases H5 (w1 H6); clear H3 H5; cases H6 (H7 H8); clear H6;
+ exists; [apply w1] split [2: assumption] constructor 1; [assumption]
+ exists; [apply w] split; assumption
+ | cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
+ cases H3 (H4 H5); cases H4 (w1 H6); clear H3 H4; cases H6 (w2 H7); clear H6;
+ cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
+ assumption]
+qed.
+
projects / helm.git / commitdiff