include "cprop_connectives.ma".
+notation "hvbox(a break = \sub \ID b)" non associative with precedence 45
+for @{ 'eqID $a $b }.
+
+notation > "hvbox(a break =_\ID b)" non associative with precedence 45
+for @{ cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? $a $b }.
+
+interpretation "ID eq" 'eqID x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y).
+
record equivalence_relation (A:Type0) : Type1 ≝
{ eq_rel:2> A → A → CProp0;
refl: reflexive ? eq_rel;
}.
-interpretation "setoid3 eq" 'eq x y = (eq_rel3 _ (eq3 _) x y).
-interpretation "setoid2 eq" 'eq x y = (eq_rel2 _ (eq2 _) x y).
-interpretation "setoid1 eq" 'eq x y = (eq_rel1 _ (eq1 _) x y).
-interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
-
-notation < "hvbox(a break = \sub 1 b)" non associative with precedence 45
-for @{ 'eq1 $a $b }.
-
-notation < "hvbox(a break = \sub 2 b)" non associative with precedence 45
-for @{ 'eq2 $a $b }.
-
-notation < "hvbox(a break = \sub 3 b)" non associative with precedence 45
-for @{ 'eq3 $a $b }.
+interpretation "setoid3 eq" 'eq t x y = (eq_rel3 _ (eq3 t) x y).
+interpretation "setoid2 eq" 'eq t x y = (eq_rel2 _ (eq2 t) x y).
+interpretation "setoid1 eq" 'eq t x y = (eq_rel1 _ (eq1 t) x y).
+interpretation "setoid eq" 'eq t x y = (eq_rel _ (eq t) x y).
notation > "hvbox(a break =_12 b)" non associative with precedence 45
for @{ eq_rel2 (carr2 (setoid2_of_setoid1 ?)) (eq2 (setoid2_of_setoid1 ?)) $a $b }.
notation > "hvbox(a break =_3 b)" non associative with precedence 45
for @{ eq_rel3 ? (eq3 ?) $a $b }.
-interpretation "setoid3 eq explicit" 'eq3 x y = (eq_rel3 _ (eq3 _) x y).
-interpretation "setoid2 eq explicit" 'eq2 x y = (eq_rel2 _ (eq2 _) x y).
-interpretation "setoid1 eq explicit" 'eq1 x y = (eq_rel1 _ (eq1 _) x y).
-
interpretation "setoid3 symmetry" 'invert r = (sym3 ____ r).
interpretation "setoid2 symmetry" 'invert r = (sym2 ____ r).
interpretation "setoid1 symmetry" 'invert r = (sym1 ____ r).
prefer coercion Type_OF_setoid. (* we prefer the lower carrier projection *)
prefer coercion Type_OF_objs1.
-interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
\ No newline at end of file
+interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
interpretation "logical iff" 'iff x y = (Iff x y).
interpretation "logical iff type1" 'iff1 x y = (Iff1 x y).
+inductive exT22 (A:Type2) (P:A→CProp2) : CProp2 ≝
+ ex_introT22: ∀w:A. P w → exT22 A P.
+
+interpretation "CProp2 exists" 'exists \eta.x = (exT22 _ x).
+
+definition pi1exT22 ≝ λA,P.λx:exT22 A P.match x with [ex_introT22 x _ ⇒ x].
+definition pi2exT22 ≝
+ λA,P.λx:exT22 A P.match x return λx.P (pi1exT22 ?? x) with [ex_introT22 _ p ⇒ p].
+
+interpretation "exT22 \fst" 'pi1 = (pi1exT22 _ _).
+interpretation "exT22 \snd" 'pi2 = (pi2exT22 _ _).
+interpretation "exT22 \fst a" 'pi1a x = (pi1exT22 _ _ x).
+interpretation "exT22 \snd a" 'pi2a x = (pi2exT22 _ _ x).
+interpretation "exT22 \fst b" 'pi1b x y = (pi1exT22 _ _ x y).
+interpretation "exT22 \snd b" 'pi2b x y = (pi2exT22 _ _ x y).
+
inductive exT (A:Type0) (P:A→CProp0) : CProp0 ≝
ex_introT: ∀w:A. P w → exT A P.
λA,P.λx:exT A P.match x return λx.P (pi1exT ?? x) with [ex_introT _ p ⇒ p].
interpretation "exT \fst" 'pi1 = (pi1exT _ _).
-interpretation "exT \fst" 'pi1a x = (pi1exT _ _ x).
-interpretation "exT \fst" 'pi1b x y = (pi1exT _ _ x y).
+interpretation "exT \fst a" 'pi1a x = (pi1exT _ _ x).
+interpretation "exT \fst b" 'pi1b x y = (pi1exT _ _ x y).
interpretation "exT \snd" 'pi2 = (pi2exT _ _).
-interpretation "exT \snd" 'pi2a x = (pi2exT _ _ x).
-interpretation "exT \snd" 'pi2b x y = (pi2exT _ _ x y).
+interpretation "exT \snd a" 'pi2a x = (pi2exT _ _ x).
+interpretation "exT \snd b" 'pi2b x y = (pi2exT _ _ x y).
inductive exT23 (A:Type0) (P:A→CProp0) (Q:A→CProp0) (R:A→A→CProp0) : CProp0 ≝
ex_introT23: ∀w,p:A. P w → Q p → R w p → exT23 A P Q R.
interpretation "exT2 \fst" 'pi1 = (pi1exT23 _ _ _ _).
interpretation "exT2 \snd" 'pi2 = (pi2exT23 _ _ _ _).
-interpretation "exT2 \fst" 'pi1a x = (pi1exT23 _ _ _ _ x).
-interpretation "exT2 \snd" 'pi2a x = (pi2exT23 _ _ _ _ x).
-interpretation "exT2 \fst" 'pi1b x y = (pi1exT23 _ _ _ _ x y).
-interpretation "exT2 \snd" 'pi2b x y = (pi2exT23 _ _ _ _ x y).
+interpretation "exT2 \fst a" 'pi1a x = (pi1exT23 _ _ _ _ x).
+interpretation "exT2 \snd a" 'pi2a x = (pi2exT23 _ _ _ _ x).
+interpretation "exT2 \fst b" 'pi1b x y = (pi1exT23 _ _ _ _ x y).
+interpretation "exT2 \snd b" 'pi2b x y = (pi2exT23 _ _ _ _ x y).
inductive exT2 (A:Type0) (P,Q:A→CProp0) : CProp0 ≝
ex_introT2: ∀w:A. P w → Q w → exT2 A P Q.
-inductive exT22 (A:Type2) (P:A→CProp2) : CProp2 ≝
- ex_introT22: ∀w:A. P w → exT22 A P.
definition Not : CProp0 → Prop ≝ λx:CProp.x → False.
notation "hvbox (r \sub \c)" with precedence 90 for @{'concr_rel $r}.
notation "hvbox (r \sub \f)" with precedence 90 for @{'form_rel $r}.
+
+notation < "mstyle color #ff0000 (…)" non associative with precedence 90
+for @{'hide}.
+
+definition hide ≝ λA:Type.λx:A.x.
+
+interpretation "hide" 'hide = (hide _ _).
(* *)
(**************************************************************************)
+include "notation.ma".
include "o-basic_pairs.ma".
include "o-basic_topologies.ma".
[ apply (Oform t);
| apply (□_t ∘ Ext⎽t);
| apply (◊_t ∘ Rest⎽t);
- | intros 2; split; intro;
+ | apply hide; intros 2; split; intro;
[ change with ((⊩) \sup ⎻* ((⊩) \sup ⎻ U) ≤ (⊩) \sup ⎻* ((⊩) \sup ⎻ V));
apply (. (#‡(lemma_10_4_a ?? (⊩) V)^-1));
apply f_minus_star_image_monotone;
| change with (U ≤ (⊩)⎻* ((⊩)⎻ U));
apply (. (or_prop2 : ?) ^ -1);
apply oa_leq_refl; ]]
- | intros 2; split; intro;
+ | apply hide; intros 2; split; intro;
[ change with (◊_t ((⊩) \sup * U) ≤ ◊_t ((⊩) \sup * V));
apply (. ((lemma_10_4_b ?? (⊩) U)^-1)‡#);
apply (f_image_monotone ?? (⊩) ? ((⊩)* V));
| change with ((⊩) ((⊩)* V) ≤ V);
apply (. (or_prop1 : ?));
apply oa_leq_refl; ]]
- | intros;
+ | apply hide; intros;
apply (.= (oa_overlap_sym' : ?));
change with ((◊_t ((⊩)* V) >< (⊩)⎻* ((⊩)⎻ U)) = (U >< (◊_t ((⊩)* V))));
apply (.= (or_prop3 ?? (⊩) ((⊩)* V) ?));
intros (BP1 BP2 t);
constructor 1;
[ apply (t \sub \f);
- | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+ | apply hide; unfold o_basic_topology_of_o_basic_pair; simplify; intros;
apply sym1;
apply (.= †(†e));
change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f ∘ (⊩)) ((⊩)* U));
change in ⊢ (? ? ? % ?) with (t \sub \f (((⊩) ∘ (⊩)* ) U));
change in e with (U=((⊩)∘(⊩ \sub BP1) \sup * ) U);
apply (†e^-1);
- | unfold o_basic_topology_of_o_basic_pair; simplify; intros;
+ | apply hide; unfold o_basic_topology_of_o_basic_pair; simplify; intros;
apply sym1;
apply (.= †(†e));
change in ⊢ (? ? ? (? ? ? ? %) ?) with ((t \sub \f⎻* ∘ (⊩)⎻* ) ((⊩)⎻ U));
apply (†e^-1);]
qed.
+
+definition OR : carr3 (arrows3 CAT2 OBP BTop).
+constructor 1;
+[ apply o_basic_topology_of_o_basic_pair;
+| intros; constructor 1;
+ [ apply o_continuous_relation_of_o_relation_pair;
+ | apply hide;
+ intros; whd; unfold o_continuous_relation_of_o_relation_pair; simplify;;
+ change with ((a \sub \f ⎻* ∘ A (o_basic_topology_of_o_basic_pair S)) =
+ (a' \sub \f ⎻*∘A (o_basic_topology_of_o_basic_pair S)));
+ whd in e; cases e; clear e e2 e3 e4;
+ change in ⊢ (? ? ? (? ? ? ? ? % ?) ?) with ((⊩\sub S)⎻* ∘ (⊩\sub S)⎻);
+ apply (.= (comp_assoc2 ? ???? ?? a\sub\f⎻* ));
+ change in ⊢ (? ? ? (? ? ? ? ? ? %) ?) with (a\sub\f ∘ ⊩\sub S)⎻*;
+ apply (.= #‡†(Ocommute:?)^-1);
+ apply (.= #‡e1);
+ change in ⊢ (? ? ? (? ? ? ? ? ? %) ?) with (⊩\sub T ∘ a'\sub\c)⎻*;
+ apply (.= #‡†(Ocommute:?));
+ change in ⊢ (? ? ? (? ? ? ? ? ? %) ?) with (a'\sub\f⎻* ∘ (⊩\sub S)⎻* );
+ apply (.= (comp_assoc2 ? ???? ?? a'\sub\f⎻* )^-1);
+ apply refl2;]
+| intros 2 (o a); apply rule #;
+| intros 6; apply refl1;]
+qed.
+
+(*
+axiom DDD : False.
+
+definition sigma_equivalence_relation2:
+ ∀C2:CAT2.∀Q.∀X,Y:exT22 ? (λy:C2.Q y).∀P.
+ equivalence_relation2 (exT22 ? (λf:arrows2 C2 (\fst X) (\fst Y).P f)).
+intros; constructor 1;
+ [ intros(F G); apply (\fst F =_2 \fst G);
+ | intro; apply refl2;
+ | intros 3; apply sym2; assumption;
+ | intros 5; apply (trans2 ?? ??? x1 x2);]
+qed.
+
+definition Apply : ∀C1,C2: CAT2.arrows3 CAT2 C1 C2 → CAT2.
+intros (C1 C2 F);
+constructor 1;
+[ apply (exT22 ? (λx:C2.exT22 ? (λy:C1.map_objs2 ?? F y =_\ID x)));
+| intros (X Y); constructor 1;
+ [ apply (exT22 ? (λf:arrows2 C2 (\fst X) (\fst Y).
+ exT22 ? (λg:arrows2 C1 (\fst (\snd X)) (\fst (\snd Y)).
+ ? (map_arrows2 ?? F ?? g) = f)));
+ intro; apply hide; clear g f; cases X in c; cases Y; cases x; cases x1; clear X Y x x1;
+ simplify; cases H; cases H1; intros; assumption;
+ | apply sigma_equivalence_relation2;]
+| intro o; constructor 1;
+ [ apply (id2 C2 (\fst o))
+ | exists[apply (id2 C1 (\fst (\snd o)))]
+ cases o; cases x; cases H; unfold hide; simplify;
+ apply (respects_id2 ?? F);]
+| intros (o1 o2 o3); constructor 1;
+ [ intros (f g); whd in f g; constructor 1;
+ [ apply (comp2 C2 (\fst o1) (\fst o2) (\fst o3) (\fst f) (\fst g));
+ | exists[apply (comp2 C1 (\fst (\snd o1)) (\fst (\snd o2)) (\fst (\snd o3)) (\fst (\snd f)) (\fst (\snd g)))]
+ cases o1; cases x; cases H;
+
(* scrivo gli statement qua cosi' verra' un conflitto :-)
1. definire il funtore OR
| i morfismi i morfismi di C1 mappati da F
| ....
]
-
+
+ E : objs CATS === Σx.∃y. F y = x
+
Quindi (Apply C1 C2 F) (che usando da ora in avanti una coercion
scrivero' (F C1) ) e' l'immagine di C1 tramite F ed e'
una sottocategoria di C2 (qualcosa da dimostare qui??? vedi sotto
con Giovanni
*)
+
projects / helm.git / commitdiff