record basic_topology: Type ≝
{ carrbt:> OA;
- A: arrows1 SET (oa_P carrbt) (oa_P carrbt);
- J: arrows1 SET (oa_P carrbt) (oa_P carrbt);
+ 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)
}.
+lemma hint: OA → objs2 OA.
+ intro; apply t;
+qed.
+coercion hint.
+
record continuous_relation (S,T: basic_topology) : Type ≝
- { cont_rel:> arrows1 ? S T;
+ { cont_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)
}.
-definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
+definition continuous_relation_setoid: basic_topology → basic_topology → setoid2.
intros (S T); constructor 1;
[ apply (continuous_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 ?));
- | simplify; intros; apply refl1;
- | simplify; intros; apply sym1; apply H
- | simplify; intros; apply trans1; [2: apply H |3: apply H1; |1: skip]]]
+ | 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 → arrows1 ? S T ≝ cont_rel.
+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.
- continuous_relation_setoid S T → unary_morphism (oa_P (carrbt S)) (oa_P (carrbt T)).
+ carr2 (continuous_relation_setoid S T) → ORelation_setoid (carrbt S) (carrbt T).
intros; apply rule cont_rel; apply c;
qed.
coercion cont_rel''.
+
(*
theorem continuous_relation_eq':
∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
qed.
*)
+
+axiom daemon: False.
definition continuous_relation_comp:
∀o1,o2,o3.
continuous_relation_setoid o1 o2 →
continuous_relation_setoid o2 o3 →
continuous_relation_setoid o1 o3.
intros (o1 o2 o3 r s); constructor 1;
- [ apply (s ∘ r)
+ [ apply (s ∘ r);
| intros;
apply sym1;
change in match ((s ∘ r) U) with (s (r U));
- (*BAD*) unfold FunClass_1_OF_carr1;
- apply (.= ((reduced : ?)\sup -1));
+ (*<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 refl1]
| intros;
- apply sym;
+ apply sym1;
change in match ((s ∘ r)⎻* U) with (s⎻* (r⎻* U));
apply (.= (saturated : ?)\sup -1);
- [ apply (.= (saturated : ?)); [ assumption | apply refl ]
- | apply refl]]
+ [ apply (.= (saturated : ?)); [ assumption | apply refl1 ]
+ | apply refl1]]
qed.
-definition BTop: category1.
+definition BTop: category2.
constructor 1;
[ apply basic_topology
| apply continuous_relation_setoid
| intro; constructor 1;
- [ apply id1
- | intros; apply H;
- | intros; apply H;]
+ [ apply id2
+ | intros; apply e;
+ | intros; apply e;]
| intros; constructor 1;
[ apply continuous_relation_comp;
- | intros; simplify; (*intro x; simplify;*)
+ | intros; simplify;
+ change with ((b⎻* ∘ a⎻* ) ∘ A o1 = ((b'⎻* ∘ a'⎻* ) ∘ A o1));
change with (b⎻* ∘ (a⎻* ∘ A o1) = b'⎻* ∘ (a'⎻* ∘ A o1));
- change in H with (a⎻* ∘ A o1 = a'⎻* ∘ A o1);
- change in H1 with (b⎻* ∘ A o2 = b'⎻* ∘ A o2);
- apply (.= H‡#);
+ change in e with (a⎻* ∘ A o1 = a'⎻* ∘ A o1);
+ change in e1 with (b⎻* ∘ A o2 = b'⎻* ∘ A o2);
+ apply (.= e‡#);
intro x;
-
- change with (eq1 (oa_P (carrbt o3)) (b⎻* (a'⎻* (A o1 x))) (b'⎻*(a'⎻* (A o1 x))));
- lapply (saturated o1 o2 a' (A o1 x):?) as X;
- [ apply ((saturation_idempotent ?? (A_is_saturation o1) x)^-1) ]
- change in X with (eq1 (oa_P (carrbt o2)) (a'⎻* (A o1 x)) (A o2 (a'⎻* (A o1 x))));
- unfold uncurry_arrows;
- apply (.= †X); whd in H1;
- lapply (H1 (a'⎻* (A o1 x))) as X1;
- change in X1 with (eq1 (oa_P (carrbt o3)) (b⎻* (A o2 (a'⎻* (A o1 x)))) (b'⎻* (A o2 (a' \sup ⎻* (A o1 x)))));
- apply (.= X1);
- unfold uncurry_arrows;
- apply (†(X\sup -1));]
+ 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);]]
| intros; simplify;
change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ A o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ A o1));
apply rule (#‡ASSOC1\sup -1);
| intros; simplify;
- change with ((a⎻* ∘ (id1 ? o1)⎻* ) ∘ A o1 = a⎻* ∘ A o1);
- apply (#‡(id_neutral_right1 : ?));
+ change with ((a⎻* ∘ (id2 ? o1)⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ apply (#‡(id_neutral_right2 : ?));
| intros; simplify;
- change with (((id1 ? o2)⎻* ∘ a⎻* ) ∘ A o1 = a⎻* ∘ A o1);
- apply (#‡(id_neutral_left1 : ?));]
+ change with (((id2 ? o2)⎻* ∘ a⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ apply (#‡(id_neutral_left2 : ?));]
qed.
(*
projects / helm.git / blobdiff