| cases daemon (*apply o_relation_topology_of_relation_topology_morphism_respects_comp*);]
qed.
-(* FIXME
-alias symbol "eq" (instance 2) = "setoid1 eq".
-alias symbol "eq" (instance 1) = "setoid2 eq".
theorem BTop_to_OBTop_faithful: faithful2 ?? BTop_to_OBTop.
- intros 5;
- whd in e; unfold BTop_to_OBTop in e; simplify in e;
- change in match (oA ?) in e with (A o1);
- whd in f g;
- alias symbol "OR_f_minus_star" (instance 1) = "relation f⎻*".
- change in match ((o_continuous_relation_of_continuous_relation_morphism o1 o2 f)⎻* ) in e with ((foo ?? f)⎻* );
- change in match ((o_continuous_relation_of_continuous_relation_morphism o1 o2 g)⎻* ) in e with ((foo ?? g)⎻* );
- whd; whd in o1 o2;
- intro b;
- alias symbol "OR_f_minus" (instance 1) = "relation f⎻".
- letin fb ≝ ((ext ?? f) b);
- lapply (e fb);
- whd in Hletin:(? ? ? % %);
- cases (Hletin); simplify in s s1;
- split;
- [2: intro; simplify;
- lapply depth=0 (s b); intro; apply (Hletin1 ? a ?)
- [ 2: whd in f1;
- change in Hletin with ((foo ?? f)⎻*
-
- alias symbol "OR_f_minus_star" (instance 4) = "relation f⎻*".
- alias symbol "OR_f_minus_star" (instance 4) = "relation f⎻*".
-change in e with
- (comp2 SET1 (Ω^o1) (Ω^o1) (Ω^o2) (A o1) (foo o1 o2 f)⎻* =_1
- comp2 SET1 (Ω^o1) (Ω^o1) (Ω^o2) (A o1) (foo o1 o2 g)⎻*
- );
-
-
-
- change in e with (comp1 SET (Ω^o1) ?? (A o1) (foo o1 o2 f)⎻* = ((foo o1 o2 g)⎻* ∘ A o1));
- unfold o_continuous_relation_of_continuous_relation_morphism in e;
- unfold o_continuous_relation_of_continuous_relation in e;
- simplify in e;
+ intros 5; apply (continuous_relation_eq_inv' o1 o2 f g); apply e;
qed.
include "formal_topology/notation.ma".
-theorem BTop_to_OBTop_full:
- ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BTop_to_OBTop S T g = f).
- intros;
+theorem BTop_to_OBTop_full: full2 ?? BTop_to_OBTop.
+ intros 3 (S T);
cases (POW_full (carrbt S) (carrbt T) (Ocont_rel ?? f)) (g Hg);
- exists[
+ (* cases Hg; *)
+ exists [
constructor 1;
[ apply g
- | apply hide; intros; lapply (Oreduced ?? f ? e);
+ | unfold image_coercion; cases daemon (*apply hide; intros; lapply (Oreduced ?? f ? e); unfold image_coercion;
cases Hg; lapply (e3 U) as K; apply (.= K);
- apply (.= Hletin); apply rule (†(K^-1));
- | apply hide; intros; lapply (Osaturated ?? f ? e);
+ apply (.= Hletin); apply rule (†(K^-1)); *)
+ | cases daemon (* apply hide; intros; lapply (Osaturated ?? f ? e);
cases Hg; lapply (e1 U) as K; apply (.= K);
- apply (.= Hletin); apply rule (†(K^-1));
+ apply (.= Hletin); apply rule (†(K^-1)); *)
]
| simplify; unfold BTop_to_OBTop; simplify;
+ cases Hg; unfold o_continuous_relation_of_continuous_relation_morphism;
+ simplify;
+ change with ((orelation_of_relation ?? g)⎻* ∘ oA (o_basic_topology_of_basic_topology S) =
+ f⎻* ∘ oA (o_basic_topology_of_basic_topology S));
+
+
+ change with (g⎻* ∘ oA (o_basic_topology_of_basic_topology S) =
+ f⎻* ∘ oA (o_basic_topology_of_basic_topology S));
+ apply sym2; whd in T;
+ intro;
+ apply trans2; [2: apply sym2; [2: apply Hg;
+
+ whd in ⊢ (?(??%%)???);
+ apply (.= Hg^-1);
unfold o_continuous_relation_of_continuous_relation_morphism; simplify;
- cases Hg; whd; simplify; intro;
+ intro; simplify;
+ unfold image_coercion; cases Hg; whd; simplify; intro; simplify;
qed.
*)