record relation_pair (BP1,BP2: basic_pair): Type1 ≝ {
concr_rel: (concr BP1) ⇒_\r1 (concr BP2); form_rel: (form BP1) ⇒_\r1 (form BP2);
- commute: ⊩ ∘ concr_rel =_1 form_rel ∘ ⊩
+ commute: comp1 REL ??? concr_rel (rel ?) =_1 form_rel ∘ ⊩
}.
interpretation "concrete relation" 'concr_rel r = (concr_rel ?? r).
∀P,Q. relation_pair_setoid P Q → relation_pair P Q ≝ λP,Q,x.x.
coercion relation_pair_of_relation_pair_setoid.
+alias symbol "compose" (instance 1) = "category1 composition".
lemma eq_to_eq':
- ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r =_1 r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
+ ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r =_1 r' → r \sub\f ∘ ⊩ =_1 r'\sub\f ∘ ⊩.
intros 5 (o1 o2 r r' H);
apply (.= (commute ?? r)^-1);
change in H with (⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
| lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
apply (.= H);
- apply (H1 \sup -1);]
+ apply (H1^-1);]
qed.
lemma relation_pair_composition:
apply (.= ASSOC ^ -1);
apply (.= e‡#);
apply (.= ASSOC);
- apply (.= #‡(commute ?? b')\sup -1);
+ apply (.= #‡(commute ?? b')^-1);
apply (ASSOC ^ -1);
qed.
(*
definition BTop_of_BP: functor1 BP BTop.
- lapply OR as F;
constructor 1;
[ apply basic_topology_of_basic_pair
| intros; constructor 1 [ apply continuous_relation_of_relation_pair; ]
| simplify; intro;
]
qed.
+
+lemma BBBB_faithful : failthful2 ?? VVV
+FIXME
*)
\ No newline at end of file
compatibility: ∀U,V. (A U ≬ J V) =_1 (U ≬ J V)
}.
+definition foo : ∀o1,o2:REL.carr1 (o1 ⇒_\r1 o2) → carr2 (setoid2_of_setoid1 (o1 ⇒_\r1 o2)) ≝ λo1,o2,x.x.
+
record continuous_relation (S,T: basic_topology) : Type1 ≝
- { cont_rel:> arrows1 ? S T;
- reduced: ∀U. U = J ? U → image ?? cont_rel U = J ? (image ?? cont_rel U);
- saturated: ∀U. U = A ? U → minus_star_image ?? cont_rel U = A ? (minus_star_image ?? cont_rel U)
+ { cont_rel:> S ⇒_\r1 T;
+ reduced: ∀U. U =_1 J ? U → image_coercion ?? cont_rel U =_1 J ? (image_coercion ?? cont_rel U);
+ saturated: ∀U. U =_1 A ? U → (foo ?? cont_rel)⎻* U = _1A ? ((foo ?? cont_rel)⎻* U)
}.
definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
axiom continuous_relation_eq':
∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
- a = a' → ∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X).
+ a = a' → ∀X.(foo ?? a)⎻* (A o1 X) = (foo ?? a')⎻* (A o1 X).
(*
intros; split; intro; unfold minus_star_image; simplify; intros;
[ cut (ext ?? a a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
axiom continuous_relation_eq_inv':
∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
- (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → a=a'.
+ (∀X.(foo ?? a)⎻* (A o1 X) = (foo ?? a')⎻* (A o1 X)) → a=a'.
(* intros 6;
cut (∀a,a': continuous_relation_setoid o1 o2.
(∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) →
continuous_relation_setoid o2 o3 →
continuous_relation_setoid o1 o3.
intros (o1 o2 o3 r s); constructor 1;
- [ apply (s ∘ r)
+ [ alias symbol "compose" (instance 1) = "category1 composition".
+apply (s ∘ r)
| intros;
- apply sym1;
+ apply sym1;
+ (*change in ⊢ (? ? ? (? ? ? ? %) ?) with (image_coercion ?? (s ∘ r) U);*)
apply (.= †(image_comp ??????));
- apply (.= (reduced ?????)\sup -1);
+ apply (.= (reduced ?? s (image_coercion ?? r U) ?)^-1);
[ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
- | apply (.= (image_comp ??????)\sup -1);
+ | change in ⊢ (? ? ? % ?) with ((image_coercion ?? s ∘ image_coercion ?? r) U);
+ apply (.= (image_comp ??????)^-1);
apply refl1]
| intros;
- apply sym1;
+ apply sym1; unfold foo;
apply (.= †(minus_star_image_comp ??????));
- apply (.= (saturated ?????)\sup -1);
+ apply (.= (saturated ?? s ((foo ?? r)⎻* U) ?)^-1);
[ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
- | apply (.= (minus_star_image_comp ??????)\sup -1);
+ | change in ⊢ (? ? ? % ?) with (((foo ?? s)⎻* ∘ (foo ?? r)⎻* ) U);
+ apply (.= (minus_star_image_comp ??????)^-1);
apply refl1]]
qed.
| intros; simplify; intro x; simplify;
lapply depth=0 (continuous_relation_eq' ???? e) as H';
lapply depth=0 (continuous_relation_eq' ???? e1) as H1';
- letin K ≝ (λX.H1' (minus_star_image ?? a (A ? X))); clearbody K;
+ letin K ≝ (λX.H1' (minus_star_image ?? (foo ?? a) (A ? X))); clearbody K;
cut (∀X:Ω \sup o1.
- minus_star_image o2 o3 b (A o2 (minus_star_image o1 o2 a (A o1 X)))
- = minus_star_image o2 o3 b' (A o2 (minus_star_image o1 o2 a' (A o1 X))));
- [2: intro; apply sym1; apply (.= #‡(†((H' ?)\sup -1))); apply sym1; apply (K X);]
+ minus_star_image o2 o3 (foo ?? b) (A o2 (minus_star_image o1 o2 (foo ?? a) (A o1 X)))
+ =_1 minus_star_image o2 o3 (foo ?? b') (A o2 (minus_star_image o1 o2 (foo ?? a') (A o1 X))));
+ [2: intro; apply sym1;
+ apply (.= (†(†((H' X)^-1)))); apply sym1; apply (K X);]
clear K H' H1';
alias symbol "compose" (instance 1) = "category1 composition".
+alias symbol "compose" (instance 1) = "category1 composition".
+alias symbol "compose" (instance 1) = "category1 composition".
cut (∀X:Ω^o1.
- minus_star_image ?? (b ∘ a) (A o1 X) =_1 minus_star_image ?? (b'∘a') (A o1 X));
- [2: intro;
+ minus_star_image ?? (foo ?? (b ∘ a)) (A o1 X) =_1 minus_star_image ?? (foo ?? (b'∘a')) (A o1 X));
+ [2: intro; unfold foo;
apply (.= (minus_star_image_comp ??????));
- apply (.= #‡(saturated ?????));
- [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
+ change in ⊢ (? ? ? % ?) with ((foo ?? b)⎻* ((foo ?? a)⎻* (A o1 X)));
+ apply (.= †(saturated ?????));
+ [ apply ((saturation_idempotent ????)^-1); apply A_is_saturation ]
apply sym1;
apply (.= (minus_star_image_comp ??????));
- apply (.= #‡(saturated ?????));
- [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
- apply ((Hcut X) \sup -1)]
+ change in ⊢ (? ? ? % ?) with ((foo ?? b')⎻* ((foo ?? a')⎻* (A o1 X)));
+ apply (.= †(saturated ?????));
+ [ apply ((saturation_idempotent ????)^-1); apply A_is_saturation ]
+ apply ((Hcut X)^-1)]
clear Hcut; generalize in match x; clear x;
apply (continuous_relation_eq_inv');
apply Hcut1;]
| 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:
- ∀S,T.∀f,g:arrows2 (category2_of_category1 BTop) S T.
- map_arrows2 ?? BTop_to_OBTop ?? f = map_arrows2 ?? BTop_to_OBTop ?? g → f=g.
- intros; change with (∀b.A ? (ext ?? f b) = A ? (ext ?? g b));
- apply (POW_faithful);
- apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) f \sub \c (⊩ \sub T));
- apply sym2;
- apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) g \sub \c (⊩ \sub T));
- apply sym2;
- apply e;
+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;
qed.
include "formal_topology/notation.ma".
λA,B:CAT2.λF:carr3 (arrows3 CAT2 A B).
∀o1,o2:A.∀f,g:arrows2 A o1 o2.F⎽⇒ f =_2 F⎽⇒ g → f =_2 g.
+
+notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
+notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
+
+notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
+notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
+
+notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
+notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
qed.
coercion arrows1_of_ORelation_setoid.
-notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
-notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
-
-notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
-notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
-
-notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
-notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
-
interpretation "o-relation f⎻*" 'OR_f_minus_star r = (fun12 ?? (or_f_minus_star ? ?) r).
interpretation "o-relation f⎻" 'OR_f_minus r = (fun12 ?? (or_f_minus ? ?) r).
interpretation "o-relation f*" 'OR_f_star r = (fun12 ?? (or_f_star ? ?) r).
definition or_prop1 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
- (F p ≤ q) = (p ≤ F* q).
+ (F p ≤ q) =_1 (p ≤ F* q).
intros; apply (or_prop1_ ?? F p q);
qed.
*)
(* the same as ⋄ for a basic pair *)
-definition image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
+definition image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^U ⇒_2 Ω^V).
intros; constructor 1;
- [ apply (λr:U ⇒_\r1 V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
- intros; simplify; split; intro; cases e1; exists [1,3: apply w]
+ [ intro r; constructor 1;
+ [ apply (λS: Ω^U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
+ intros; simplify; split; intro; cases e1; exists [1,3: apply w]
[ apply (. (#‡e^-1)‡#); assumption
| apply (. (#‡e)‡#); assumption]
- | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
- [ apply (. #‡(#‡e1^-1)); cases x; split; try assumption;
- apply (if ?? (e ??)); assumption
- | apply (. #‡(#‡e1)); cases x; split; try assumption;
- apply (if ?? (e ^ -1 ??)); assumption]]
+ | intros; split;
+ [ intro y; simplify; intro yA; cases yA; exists; [ apply w ];
+ apply (. #‡(#‡e^-1)); assumption;
+ | intro y; simplify; intro yA; cases yA; exists; [ apply w ];
+ apply (. #‡(#‡e)); assumption;]]
+ | simplify; intros; intro y; simplify; split; simplify; intros (b H); cases H;
+ exists; [1,3: apply w]; cases x; split; try assumption;
+ [ apply (if ?? (e ??)); | apply (fi ?? (e ??)); ] assumption;]
qed.
(* the same as □ for a basic pair *)
-definition minus_star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^U ⇒_1 Ω^V.
- intros; constructor 1;
- [ apply (λr:U ⇒_\r1 V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
- intros; simplify; split; intros; apply f;
- [ apply (. #‡e); assumption
- | apply (. #‡e ^ -1); assumption]
- | intros; split; simplify; intros; [ apply (. #‡e1^ -1); | apply (. #‡e1 )]
- apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
+definition minus_star_image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^U ⇒_2 Ω^V).
+ intros; constructor 1; intros;
+ [ constructor 1;
+ [ apply (λS: Ω^U. {y | ∀x:U. x ♮c y → x ∈ S});
+ intros; simplify; split; intros; apply f;
+ [ apply (. #‡e); | apply (. #‡e ^ -1)] assumption;
+ | intros; split; intro; simplify; intros;
+ [ apply (. #‡e^-1);| apply (. #‡e); ] apply f; assumption;]
+ | intros; intro; simplify; split; simplify; intros; apply f;
+ [ apply (. (e x a2)); assumption | apply (. (e^-1 x a2)); assumption]]
qed.
(* the same as Rest for a basic pair *)
-definition star_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
+definition star_image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^V ⇒_2 Ω^U).
intros; constructor 1;
- [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
- intros; simplify; split; intros; apply f;
- [ apply (. e ‡#); assumption
- | apply (. e^ -1‡#); assumption]
- | intros; split; simplify; intros; [ apply (. #‡e1 ^ -1); | apply (. #‡e1)]
- apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
+ [ intro r; constructor 1;
+ [ apply (λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
+ intros; simplify; split; intros; apply f;
+ [ apply (. e ‡#);| apply (. e^ -1‡#);] assumption;
+ | intros; split; simplify; intros;
+ [ apply (. #‡e^-1);| apply (. #‡e); ] apply f; assumption;]
+ | intros; intro; simplify; split; simplify; intros; apply f;
+ [ apply (. e a2 y); | apply (. e^-1 a2 y)] assumption;]
qed.
(* the same as Ext for a basic pair *)
-definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
+definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^V ⇒_2 Ω^U).
intros; constructor 1;
- [ apply (λr:U ⇒_\r1 V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
- exT ? (λy:V.x ♮r y ∧ y ∈ S) });
- intros; simplify; split; intro; cases e1; exists [1,3: apply w]
- [ apply (. (e ^ -1‡#)‡#); assumption
- | apply (. (e‡#)‡#); assumption]
- | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
- [ apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
- apply (if ?? (e ??)); assumption
- | apply (. #‡(#‡e1)); cases x; split; try assumption;
- apply (if ?? (e ^ -1 ??)); assumption]]
+ [ intro r; constructor 1;
+ [ apply (λS: Ω^V. {x | ∃y:V. x ♮r y ∧ y ∈ S }).
+ intros; simplify; split; intros; cases e1; cases x; exists; [1,3: apply w]
+ split; try assumption; [ apply (. (e^-1‡#)); | apply (. (e‡#));] assumption;
+ | intros; simplify; split; simplify; intros; cases e1; cases x;
+ exists [1,3: apply w] split; try assumption;
+ [ apply (. (#‡e^-1)); | apply (. (#‡e));] assumption]
+ | intros; intro; simplify; split; simplify; intros; cases e1; exists [1,3: apply w]
+ cases x; split; try assumption;
+ [ apply (. e^-1 a2 w); | apply (. e a2 w)] assumption;]
qed.
+interpretation "relation f⎻*" 'OR_f_minus_star r = (fun12 ?? (minus_star_image ? ?) r).
+interpretation "relation f⎻" 'OR_f_minus r = (fun12 ?? (minus_image ? ?) r).
+interpretation "relation f*" 'OR_f_star r = (fun12 ?? (star_image ? ?) r).
+
+definition image_coercion: ∀U,V:REL. (U ⇒_\r1 V) → Ω^U ⇒_2 Ω^V.
+intros (U V r Us); apply (image U V r); qed.
+coercion image_coercion.
+
(* minus_image is the same as ext *)
-theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
- intros; unfold image; simplify; split; simplify; intros;
+theorem image_id: ∀o. (id1 REL o : carr2 (Ω^o ⇒_2 Ω^o)) =_1 (id2 SET1 Ω^o).
+ intros; unfold image_coercion; unfold image; simplify;
+ whd in match (?:carr2 ?);
+ intro U; simplify; split; simplify; intros;
[ change with (a ∈ U);
- cases e; cases x; change in f with (eq1 ? w a); apply (. f^-1‡#); assumption
+ cases e; cases x; change in e1 with (w =_1 a); apply (. e1^-1‡#); assumption
| change in f with (a ∈ U);
exists; [apply a] split; [ change with (a = a); apply refl1 | assumption]]
qed.
-theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
- intros; unfold minus_star_image; simplify; split; simplify; intros;
+theorem minus_star_image_id: ∀o:REL. (fun12 ?? (minus_star_image o o) (id1 REL o) : carr2 (Ω^o ⇒_2 Ω^o)) =_1 (id2 SET1 Ω^o).
+ intros; unfold minus_star_image; simplify; intro U; simplify;
+ split; simplify; intros;
[ change with (a ∈ U); apply f; change with (a=a); apply refl1
| change in f1 with (eq1 ? x a); apply (. f1‡#); apply f]
qed.
-alias symbol "compose" (instance 2) = "category1 composition".
-theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
- intros; unfold image; simplify; split; simplify; intros; cases e; clear e; cases x;
- clear x; [ cases f; clear f; | cases f1; clear f1 ]
- exists; try assumption; cases x; clear x; split; try assumption;
- exists; try assumption; split; assumption.
+alias symbol "compose" (instance 5) = "category2 composition".
+alias symbol "compose" (instance 4) = "category1 composition".
+theorem image_comp: ∀A,B,C.∀r:B ⇒_\r1 C.∀s:A ⇒_\r1 B.
+ ((r ∘ s) : carr2 (Ω^A ⇒_2 Ω^C)) =_1 r ∘ s.
+ intros; intro U; split; intro x; (unfold image; unfold SET1; simplify);
+ intro H; cases H;
+ cases x1; [cases f|cases f1]; exists; [1,3: apply w1] cases x2; split; try assumption;
+ exists; try assumption; split; assumption;
qed.
theorem minus_star_image_comp:
- ∀A,B,C,r,s,X.
- minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
- intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
- [ apply f; exists; try assumption; split; assumption
- | change with (x ∈ X); cases f1; cases x1; apply f; assumption]
+ ∀A,B,C.∀r:B ⇒_\r1 C.∀s:A ⇒_\r1 B.
+ minus_star_image A C (r ∘ s) =_1 minus_star_image B C r ∘ (minus_star_image A B s).
+ intros; unfold minus_star_image; intro X; simplify; split; simplify; intros;
+ [ whd; intros; simplify; whd; intros; apply f; exists; try assumption; split; assumption;
+ | cases f1; cases x1; apply f; assumption]
qed.
(*
definition powerset_of_POW': ∀A.oa_P (POW' A) → Ω^A ≝ λA,x.x.
coercion powerset_of_POW'.
+definition foo : ∀o1,o2:REL.carr1 (o1 ⇒_\r1 o2) → carr2 (setoid2_of_setoid1 (o1 ⇒_\r1 o2)) ≝ λo1,o2,x.x.
+
definition orelation_of_relation: ∀o1,o2:REL. o1 ⇒_\r1 o2 → (POW' o1) ⇒_\o2 (POW' o2).
intros;
constructor 1;
- [ constructor 1;
- [ apply (λU.image ?? c U);
- | intros; apply (#‡e); ]
- | constructor 1;
- [ apply (λU.minus_star_image ?? c U);
- | intros; apply (#‡e); ]
- | constructor 1;
- [ apply (λU.star_image ?? c U);
- | intros; apply (#‡e); ]
- | constructor 1;
- [ apply (λU.minus_image ?? c U);
- | intros; apply (#‡e); ]
- | intros; split; intro;
- [ change in f with (∀a. a ∈ image ?? c p → a ∈ q);
- change with (∀a:o1. a ∈ p → a ∈ star_image ?? c q);
- intros 4; apply f; exists; [apply a] split; assumption;
- | change in f with (∀a:o1. a ∈ p → a ∈ star_image ?? c q);
- change with (∀a. a ∈ image ?? c p → a ∈ q);
- intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
+ [ apply rule c;
+ | apply rule ((foo ?? c)⎻* );
+ | apply rule ((foo ?? c)* );
+ | apply rule ((foo ?? c)⎻);
| intros; split; intro;
- [ change in f with (∀a. a ∈ minus_image ?? c p → a ∈ q);
- change with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? c q);
- intros 4; apply f; exists; [apply a] split; assumption;
- | change in f with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? c q);
- change with (∀a. a ∈ minus_image ?? c p → a ∈ q);
- intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
- | intros; split; intro; cases f; clear f;
+ [ intros 2; intros 2; apply (f y); exists[apply a] split; assumption;
+ | intros 2; change with (a ∈ q); cases f1; cases x; clear f1 x;
+ apply (f w f3); assumption; ]
+ | unfold foo; intros; split; intro;
+ [ intros 2; intros 2; apply (f x); exists [apply a] split; assumption;
+ | intros 2; change with (a ∈ q); cases f1; cases x; apply (f w f3); assumption;]
+ | intros; split; unfold foo; unfold image_coercion; simplify; intro; cases f; clear f;
[ cases x; cases x2; clear x x2; exists; [apply w1]
- [ assumption;
- | exists; [apply w] split; assumption]
+ [ assumption | exists; [apply w] split; assumption]
| cases x1; cases x2; clear x1 x2; exists; [apply w1]
[ exists; [apply w] split; assumption;
| assumption; ]]]
lemma orelation_of_relation_preserves_equality:
∀o1,o2:REL.∀t,t': o1 ⇒_\r1 o2.
t = t' → orelation_of_relation ?? t =_2 orelation_of_relation ?? t'.
- intros; split; unfold orelation_of_relation; simplify; intro; split; intro;
- simplify; whd in o1 o2;
- [ change with (a1 ∈ minus_star_image ?? t a → a1 ∈ minus_star_image ?? t' a);
- apply (. #‡(e^-1‡#));
- | change with (a1 ∈ minus_star_image ?? t' a → a1 ∈ minus_star_image ?? t a);
- apply (. #‡(e‡#));
- | change with (a1 ∈ minus_image ?? t a → a1 ∈ minus_image ?? t' a);
- apply (. #‡(e ^ -1‡#));
- | change with (a1 ∈ minus_image ?? t' a → a1 ∈ minus_image ?? t a);
- apply (. #‡(e‡#));
- | change with (a1 ∈ image ?? t a → a1 ∈ image ?? t' a);
- apply (. #‡(e ^ -1‡#));
- | change with (a1 ∈ image ?? t' a → a1 ∈ image ?? t a);
- apply (. #‡(e‡#));
- | change with (a1 ∈ star_image ?? t a → a1 ∈ star_image ?? t' a);
- apply (. #‡(e ^ -1‡#));
- | change with (a1 ∈ star_image ?? t' a → a1 ∈ star_image ?? t a);
- apply (. #‡(e‡#)); ]
+ intros; split; unfold orelation_of_relation; unfold foo; simplify;
+ change in e with (t =_2 t'); unfold image_coercion; apply (†e);
qed.
+lemma minus_image_id : ∀o:REL.(foo ?? (id1 REL o))⎻ =_1 (id2 SET1 Ω^o).
+unfold foo; intro o; intro; unfold minus_image; simplify; split; simplify; intros;
+[ cases e; cases x; change with (a1 ∈ a); change in f with (a1 =_1 w);
+ apply (. f‡#); assumption;
+| change in f with (a1 ∈ a); exists [ apply a1] split; try assumption;
+ change with (a1 =_1 a1); apply refl1;]
+qed.
+
+lemma star_image_id : ∀o:REL. (foo ?? (id1 REL o))* =_1 (id2 SET1 Ω^o).
+unfold foo; intro o; intro; unfold star_image; simplify; split; simplify; intros;
+[ change with (a1 ∈ a); apply f; change with (a1 =_1 a1); apply rule refl1;
+| change in f1 with (a1 =_1 y); apply (. f1^-1‡#); apply f;]
+qed.
+
lemma orelation_of_relation_preserves_identity:
∀o1:REL. orelation_of_relation ?? (id1 ? o1) =_2 id2 OA (POW' o1).
- intros; split; intro; split; whd; intro;
+ intros; split;
+ (unfold orelation_of_relation; unfold OA; unfold foo; simplify);
+ [ apply (minus_star_image_id o1);
+ | apply (minus_image_id o1);
+ | apply (image_id o1);
+ | apply (star_image_id o1) ]
+qed.
+
+(*
+ split; whd; intro;
[ change with ((∀x. x ♮(id1 REL o1) a1→x∈a) → a1 ∈ a); intros;
apply (f a1); change with (a1 = a1); apply refl1;
| change with (a1 ∈ a → ∀x. x ♮(id1 REL o1) a1→x∈a); intros;
| change with (a1 ∈ a → ∀y.a1 ♮(id1 REL o1) y→y∈a); intros;
change in f1 with (a1 = y); apply (. f1^-1‡#); apply f;]
qed.
+*)
(* CSC: ???? forse un uncertain mancato *)
alias symbol "eq" = "setoid2 eq".
theorem POW_faithful: faithful2 ?? POW.
intros 5; unfold POW in e; simplify in e; cases e;
unfold orelation_of_relation in e3; simplify in e3; clear e e1 e2 e4;
- intros 2; cases (e3 {(x)});
+ intros 2; simplify; unfold image_coercion in e3; cases (e3 {(x)});
split; intro; [ lapply (s y); | lapply (s1 y); ]
[2,4: exists; [1,3:apply x] split; [1,3: assumption |*: change with (x=x); apply rule #]
|*: cases Hletin; cases x1; change in f3 with (x =_1 w); apply (. f3‡#); assumption;]
qed.
+
(*
lemma currify: ∀A,B,C. (A × B ⇒_1 C) → A → (B ⇒_1 C).
intros; constructor 1; [ apply (b c); | intros; apply (#‡e);]
qed.
*)
+include "formal_topology/notation.ma".
+
theorem POW_full: full2 ?? POW.
intros 3 (S T); exists;
[ constructor 1; constructor 1;
[ apply (λx:carr S.λy:carr T. y ∈ f {(x)});
- | intros; unfold FunClass_1_OF_carr2; lapply (.= e1‡#);
+ | apply hide; intros; unfold FunClass_1_OF_carr2; lapply (.= e1‡#);
[4: apply mem; |6: apply Hletin;|1,2,3,5: skip]
lapply (#‡prop11 ?? f ?? (†e)); [6: apply Hletin; |*:skip ]]
- | whd; split; whd; intro; simplify; unfold map_arrows2; simplify;
- [ split;
+ | (split; intro; split; simplify);
+ (*
+ whd; split; whd; intro; simplify; unfold map_arrows2; simplify;
+ [ split;*)
[ change with (∀a1.(∀x. a1 ∈ (f {(x):S}) → x ∈ a) → a1 ∈ f⎻* a);
- | change with (∀a1.a1 ∈ f⎻* a → (∀x.a1 ∈ f {(x):S} → x ∈ a)); ]
- | split;
- [ change with (∀a1.(∃y:carr T. y ∈ f {(a1):S} ∧ y ∈ a) → a1 ∈ f⎻ a);
- | change with (∀a1.a1 ∈ f⎻ a → (∃y:carr T.y ∈ f {(a1):S} ∧ y ∈ a)); ]
- | split;
- [ change with (∀a1.(∃x:carr S. a1 ∈ f {(x):S} ∧ x ∈ a) → a1 ∈ f a);
- | change with (∀a1.a1 ∈. f a → (∃x:carr S. a1 ∈ f {(x):S} ∧ x ∈ a)); ]
- | split;
- [ change with (∀a1.(∀y. y ∈ f {(a1):S} → y ∈ a) → a1 ∈ f* a);
- | change with (∀a1.a1 ∈ f* a → (∀y. y ∈ f {(a1):S} → y ∈ a)); ]]
+ | change with (∀a1.a1 ∈ f⎻* a → (∀x.a1 ∈ f {(x):S} → x ∈ a));
+ | alias symbol "and" (instance 4) = "and_morphism".
+change with (∀a1.(∃y:carr T. y ∈ f {(a1):S} ∧ y ∈ a) → a1 ∈ f⎻ a);
+ | alias symbol "and" (instance 2) = "and_morphism".
+change with (∀a1.a1 ∈ f⎻ a → (∃y:carr T.y ∈ f {(a1):S} ∧ y ∈ a));
+ | alias symbol "and" (instance 3) = "and_morphism".
+change with (∀a1.(∃x:carr S. a1 ∈ f {(x):S} ∧ x ∈ a) → a1 ∈ f a);
+ | change with (∀a1.a1 ∈. f a → (∃x:carr S. a1 ∈ f {(x):S} ∧ x ∈ a));
+ | change with (∀a1.(∀y. y ∈ f {(a1):S} → y ∈ a) → a1 ∈ f* a);
+ | change with (∀a1.a1 ∈ f* a → (∀y. y ∈ f {(a1):S} → y ∈ a)); ]
[ intros; apply ((. (or_prop2 ?? f (singleton ? a1) a)^-1) ? a1);
[ intros 2; apply (f1 a2); change in f2 with (a2 ∈ f⎻ (singleton ? a1));
lapply (. (or_prop3 ?? f (singleton ? a2) (singleton ? a1)));
intros; simplify;
split; intro;
apply (.= e1);
- [ apply e | apply (e \sup -1) ]
+ [ apply e | apply (e^-1) ]
| unfold setoid1_of_setoid; simplify;
intros; split; intros 2; simplify in f ⊢ %; apply trans;
[ apply a |4: apply a'] try assumption; apply sym; assumption]
projects / helm.git / commitdiff