-(**************************************************************************)
+ (**************************************************************************)
(* ___ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
include "o-algebra.ma".
include "o-saturations.ma".
-record basic_topology: Type2 ≝
- { carrbt:> OA;
- A: carrbt ⇒ carrbt;
- J: carrbt ⇒ carrbt;
- A_is_saturation: is_saturation ? A;
- J_is_reduction: is_reduction ? J;
- compatibility: ∀U,V. (A U >< J V) = (U >< J V)
+record Obasic_topology: Type2 ≝
+ { Ocarrbt:> OA;
+ oA: Ocarrbt ⇒ Ocarrbt;
+ oJ: Ocarrbt ⇒ Ocarrbt;
+ oA_is_saturation: is_o_saturation ? oA;
+ oJ_is_reduction: is_o_reduction ? oJ;
+ Ocompatibility: ∀U,V. (oA U >< oJ V) = (U >< oJ V)
}.
-lemma hint: OA → objs2 OA.
- intro; apply t;
-qed.
-coercion hint.
-
-record continuous_relation (S,T: basic_topology) : Type2 ≝
- { cont_rel:> arrows2 OA S T;
+record Ocontinuous_relation (S,T: Obasic_topology) : Type2 ≝
+ { Ocont_rel:> arrows2 OA S T;
(* reduces uses eq1, saturated uses eq!!! *)
- reduced: ∀U. U = J ? U → cont_rel U = J ? (cont_rel U);
- saturated: ∀U. U = A ? U → cont_rel⎻* U = A ? (cont_rel⎻* U)
+ Oreduced: ∀U. U = oJ ? U → Ocont_rel U = oJ ? (Ocont_rel U);
+ Osaturated: ∀U. U = oA ? U → Ocont_rel⎻* U = oA ? (Ocont_rel⎻* U)
}.
-definition continuous_relation_setoid: basic_topology → basic_topology → setoid2.
+definition Ocontinuous_relation_setoid: Obasic_topology → Obasic_topology → setoid2.
intros (S T); constructor 1;
- [ apply (continuous_relation S T)
+ [ apply (Ocontinuous_relation S T)
| constructor 1;
- [ (*apply (λr,s:continuous_relation S T.∀b. eq1 (oa_P (carrbt S)) (A ? (r⎻ b)) (A ? (s⎻ b)));*)
- apply (λr,s:continuous_relation S T.r⎻* ∘ (A S) = s⎻* ∘ (A ?));
+ [ alias symbol "eq" = "setoid2 eq".
+ alias symbol "compose" = "category2 composition".
+ apply (λr,s:Ocontinuous_relation S T. (r⎻* ) ∘ (oA S) = (s⎻* ∘ (oA ?)));
| simplify; intros; apply refl2;
| simplify; intros; apply sym2; apply e
| simplify; intros; apply trans2; [2: apply e |3: apply e1; |1: skip]]]
qed.
-definition cont_rel': ∀S,T: basic_topology. continuous_relation_setoid S T → arrows2 ? S T ≝ cont_rel.
-
-coercion cont_rel'.
-
-definition cont_rel'':
- ∀S,T: basic_topology.
- carr2 (continuous_relation_setoid S T) → ORelation_setoid (carrbt S) (carrbt T).
- intros; apply rule cont_rel; apply c;
-qed.
-
-coercion cont_rel''.
+definition Ocontinuous_relation_of_Ocontinuous_relation_setoid:
+ ∀P,Q. Ocontinuous_relation_setoid P Q → Ocontinuous_relation P Q ≝ λP,Q,c.c.
+coercion Ocontinuous_relation_of_Ocontinuous_relation_setoid.
(*
theorem continuous_relation_eq':
qed.
*)
-axiom daemon: False.
-definition continuous_relation_comp:
+definition Ocontinuous_relation_comp:
∀o1,o2,o3.
- continuous_relation_setoid o1 o2 →
- continuous_relation_setoid o2 o3 →
- continuous_relation_setoid o1 o3.
+ Ocontinuous_relation_setoid o1 o2 →
+ Ocontinuous_relation_setoid o2 o3 →
+ Ocontinuous_relation_setoid o1 o3.
intros (o1 o2 o3 r s); constructor 1;
[ apply (s ∘ r);
| intros;
- apply sym1;
+ apply sym1;
change in match ((s ∘ r) U) with (s (r U));
- (*<BAD>*) unfold FunClass_1_OF_Type_OF_setoid2;
- unfold objs2_OF_basic_topology1; unfold hint;
- letin reduced := reduced; clearbody reduced;
- unfold uncurry_arrows in reduced ⊢ %; (*</BAD>*)
- apply (.= (reduced : ?)\sup -1);
- [ (*BAD*) change with (eq1 ? (r U) (J ? (r U)));
- (* BAD U *) apply (.= (reduced ??? U ?)); [ assumption | apply refl1 ]
+ apply (.= (Oreduced : ?)\sup -1);
+ [ apply (.= (Oreduced :?)); [ assumption | apply refl1 ]
| apply refl1]
| intros;
apply sym1;
change in match ((s ∘ r)⎻* U) with (s⎻* (r⎻* U));
- apply (.= (saturated : ?)\sup -1);
- [ apply (.= (saturated : ?)); [ assumption | apply refl1 ]
+ apply (.= (Osaturated : ?)\sup -1);
+ [ apply (.= (Osaturated : ?)); [ assumption | apply refl1 ]
| apply refl1]]
qed.
-definition BTop: category2.
+definition OBTop: category2.
constructor 1;
- [ apply basic_topology
- | apply continuous_relation_setoid
+ [ apply Obasic_topology
+ | apply Ocontinuous_relation_setoid
| intro; constructor 1;
[ apply id2
| intros; apply e;
| intros; apply e;]
| intros; constructor 1;
- [ apply continuous_relation_comp;
+ [ apply Ocontinuous_relation_comp;
| intros; simplify;
- change with ((b⎻* ∘ a⎻* ) ∘ A o1 = ((b'⎻* ∘ a'⎻* ) ∘ A o1));
- change with (b⎻* ∘ (a⎻* ∘ A o1) = b'⎻* ∘ (a'⎻* ∘ A o1));
- change in e with (a⎻* ∘ A o1 = a'⎻* ∘ A o1);
- change in e1 with (b⎻* ∘ A o2 = b'⎻* ∘ A o2);
+ change with ((b⎻* ∘ a⎻* ) ∘ oA o1 = ((b'⎻* ∘ a'⎻* ) ∘ oA o1));
+ change with (b⎻* ∘ (a⎻* ∘ oA o1) = b'⎻* ∘ (a'⎻* ∘ oA o1));
+ change in e with (a⎻* ∘ oA o1 = a'⎻* ∘ oA o1);
+ change in e1 with (b⎻* ∘ oA o2 = b'⎻* ∘ oA o2);
apply (.= e‡#);
- intro x;
- change with (b⎻* (a'⎻* (A o1 x)) = b'⎻*(a'⎻* (A o1 x)));
- alias symbol "trans" = "trans1".
- alias symbol "prop1" = "prop11".
- alias symbol "invert" = "setoid1 symmetry".
- lapply (.= †(saturated o1 o2 a' (A o1 x) : ?));
- [3: apply (b⎻* ); | 5: apply Hletin; |1,2: skip;
- |apply ((saturation_idempotent ?? (A_is_saturation o1) x)^-1); ]
- change in e1 with (∀x.b⎻* (A o2 x) = b'⎻* (A o2 x));
- apply (.= (e1 (a'⎻* (A o1 x))));
- alias symbol "invert" = "setoid1 symmetry".
- lapply (†((saturated ?? a' (A o1 x) : ?) ^ -1));
- [2: apply (b'⎻* ); |4: apply Hletin; | skip;
- |apply ((saturation_idempotent ?? (A_is_saturation o1) x)^-1);]]
+ intro x;
+ change with (eq1 ? (b⎻* (a'⎻* (oA o1 x))) (b'⎻*(a'⎻* (oA o1 x))));
+ apply (.= †(Osaturated o1 o2 a' (oA o1 x) ?)); [
+ apply ((o_saturation_idempotent ?? (oA_is_saturation o1) x)^-1);]
+ apply (.= (e1 (a'⎻* (oA o1 x))));
+ change with (eq1 ? (b'⎻* (oA o2 (a'⎻* (oA o1 x)))) (b'⎻*(a'⎻* (oA o1 x))));
+ apply (.= †(Osaturated o1 o2 a' (oA o1 x):?)^-1); [
+ apply ((o_saturation_idempotent ?? (oA_is_saturation o1) x)^-1);]
+ apply rule #;]
| intros; simplify;
- change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ A o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ A o1));
+ change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ oA o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ oA o1));
apply rule (#‡ASSOC ^ -1);
| intros; simplify;
- change with ((a⎻* ∘ (id2 ? o1)⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ change with ((a⎻* ∘ (id2 ? o1)⎻* ) ∘ oA o1 = a⎻* ∘ oA o1);
apply (#‡(id_neutral_right2 : ?));
| intros; simplify;
- change with (((id2 ? o2)⎻* ∘ a⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ change with (((id2 ? o2)⎻* ∘ a⎻* ) ∘ oA o1 = a⎻* ∘ oA o1);
apply (#‡(id_neutral_left2 : ?));]
qed.
-definition btop_carr: BTop → Type1 ≝ λo:BTop. carr1 (oa_P (carrbt o)).
-coercion btop_carr.
+definition Obasic_topology_of_OBTop: objs2 OBTop → Obasic_topology ≝ λx.x.
+coercion Obasic_topology_of_OBTop.
+
+definition Ocontinuous_relation_setoid_of_arrows2_OBTop :
+ ∀P,Q. arrows2 OBTop P Q → Ocontinuous_relation_setoid P Q ≝ λP,Q,x.x.
+coercion Ocontinuous_relation_setoid_of_arrows2_OBTop.
(*
(*CSC: unused! *)