From ccac4e720ff2a9bee8e1c9d5ba1ea6474db72572 Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Mon, 26 Jan 2009 17:12:56 +0000 Subject: [PATCH] minor fixes --- .../formal_topology/overlap/categories.ma | 31 ++++---- .../overlap/cprop_connectives.ma | 34 ++++++--- .../formal_topology/overlap/notation.ma | 7 ++ .../o-basic_pairs_to_o-basic_topologies.ma | 76 +++++++++++++++++-- 4 files changed, 114 insertions(+), 34 deletions(-) diff --git a/helm/software/matita/contribs/formal_topology/overlap/categories.ma b/helm/software/matita/contribs/formal_topology/overlap/categories.ma index 17c5f498f..3e750928d 100644 --- a/helm/software/matita/contribs/formal_topology/overlap/categories.ma +++ b/helm/software/matita/contribs/formal_topology/overlap/categories.ma @@ -14,6 +14,14 @@ 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; @@ -96,19 +104,10 @@ record setoid3: Type4 ≝ }. -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 }. @@ -121,10 +120,6 @@ for @{ eq_rel2 ? (eq2 ?) $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). @@ -467,4 +462,4 @@ coercion objs2_of_category1. 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). diff --git a/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma b/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma index 59473882d..13bb825ee 100644 --- a/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma +++ b/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma @@ -102,6 +102,22 @@ notation "hvbox(a break ⇔ b)" right associative with precedence 25 for @{'iff1 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. @@ -118,11 +134,11 @@ definition pi2exT ≝ 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. @@ -134,16 +150,14 @@ definition pi2exT23 ≝ 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. diff --git a/helm/software/matita/contribs/formal_topology/overlap/notation.ma b/helm/software/matita/contribs/formal_topology/overlap/notation.ma index 97c1a9b6d..2cb92807d 100644 --- a/helm/software/matita/contribs/formal_topology/overlap/notation.ma +++ b/helm/software/matita/contribs/formal_topology/overlap/notation.ma @@ -14,3 +14,10 @@ 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 _ _). diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma index 432025c99..bfd76f99e 100644 --- a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma +++ b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +include "notation.ma". include "o-basic_pairs.ma". include "o-basic_topologies.ma". @@ -24,7 +25,7 @@ definition o_basic_topology_of_o_basic_pair: OBP → BTop. [ 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; @@ -36,7 +37,7 @@ definition o_basic_topology_of_o_basic_pair: OBP → BTop. | 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)); @@ -48,7 +49,7 @@ definition o_basic_topology_of_o_basic_pair: OBP → BTop. | 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) ?)); @@ -63,7 +64,7 @@ definition o_continuous_relation_of_o_relation_pair: 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)); @@ -76,7 +77,7 @@ definition o_continuous_relation_of_o_relation_pair: 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)); @@ -91,6 +92,66 @@ definition o_continuous_relation_of_o_relation_pair: 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 @@ -105,7 +166,9 @@ qed. | 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 @@ -164,3 +227,4 @@ qed. con Giovanni *) + -- 2.39.2