From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Wed, 21 Jan 2009 17:52:33 +0000 (+0000)
Subject: some minor fixes
X-Git-Tag: make_still_working~4237
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=3e094922bf3fec6975fdbe6feceb509eaafe0563;p=helm.git

some minor fixes
---

diff --git a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_o-basic_pairs.ma b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_o-basic_pairs.ma
index 2aec4f7e8..d8813fdc1 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_o-basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_o-basic_pairs.ma
@@ -20,9 +20,9 @@ include "relations_to_o-algebra.ma".
 definition o_basic_pair_of_basic_pair: basic_pair → Obasic_pair.
  intro b;
  constructor 1;
-  [ apply (map_objs2 ?? SUBSETS' (concr b));
-  | apply (map_objs2 ?? SUBSETS' (form b));
-  | apply (map_arrows2 ?? SUBSETS' (concr b) (form b) (rel b)); ]
+  [ apply (map_objs2 ?? POW (concr b));
+  | apply (map_objs2 ?? POW (form b));
+  | apply (map_arrows2 ?? POW (concr b) (form b) (rel b)); ]
 qed.
 
 definition o_relation_pair_of_relation_pair:
@@ -30,12 +30,12 @@ definition o_relation_pair_of_relation_pair:
   Orelation_pair (o_basic_pair_of_basic_pair BP1) (o_basic_pair_of_basic_pair BP2).
  intros;
  constructor 1;
-  [ apply (map_arrows2 ?? SUBSETS' (concr BP1) (concr BP2) (r \sub \c));
-  | apply (map_arrows2 ?? SUBSETS' (form BP1) (form BP2) (r \sub \f));
-  | apply (.= (respects_comp2 ?? SUBSETS' (concr BP1) (concr BP2) (form BP2)  r\sub\c (⊩\sub BP2) )^-1);
-    cut (⊩ \sub BP2∘r \sub \c = r\sub\f ∘ ⊩ \sub BP1) as H;
+  [ apply (map_arrows2 ?? POW (concr BP1) (concr BP2) (r \sub \c));
+  | apply (map_arrows2 ?? POW (form BP1) (form BP2) (r \sub \f));
+  | apply (.= (respects_comp2 ?? POW (concr BP1) (concr BP2) (form BP2)  r\sub\c (⊩\sub BP2) )^-1);
+    cut ( ⊩ \sub BP2 ∘ r \sub \c =_12 r\sub\f ∘ ⊩ \sub BP1) as H;
     [ apply (.= †H);
-      apply (respects_comp2 ?? SUBSETS' (concr BP1) (form BP1) (form BP2) (⊩\sub BP1) r\sub\f);
+      apply (respects_comp2 ?? POW (concr BP1) (form BP1) (form BP2) (⊩\sub BP1) r\sub\f);
     | apply commute;]]
 qed.
 
@@ -52,9 +52,9 @@ definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 BP) OBP).
        | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
        simplify;
        apply (prop12);
-       apply (.= (respects_comp2 ?? SUBSETS' (concr S) (concr T) (form T) (a\sub\c) (⊩\sub T))^-1);
+       apply (.= (respects_comp2 ?? POW (concr S) (concr T) (form T) (a\sub\c) (⊩\sub T))^-1);
        apply sym2;
-       apply (.= (respects_comp2 ?? SUBSETS' (concr S) (concr T) (form T) (b\sub\c) (⊩\sub T))^-1);
+       apply (.= (respects_comp2 ?? POW (concr S) (concr T) (form T) (b\sub\c) (⊩\sub T))^-1);
        apply sym2;
        apply prop12;
        apply Eab;
@@ -67,7 +67,7 @@ definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 BP) OBP).
     simplify;
     apply prop12;
     apply prop22;[2,4,6,8: apply rule #;]
-    apply (respects_id2 ?? SUBSETS' (concr o)); 
+    apply (respects_id2 ?? POW (concr o)); 
   | simplify; intros; whd; split;
        [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x); 
        | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
@@ -76,33 +76,33 @@ definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 BP) OBP).
     simplify;
     apply prop12;
     apply prop22;[2,4,6,8: apply rule #;]
-    apply (respects_comp2 ?? SUBSETS' (concr o1) (concr o2) (concr o3) f1\sub\c f2\sub\c);]     
+    apply (respects_comp2 ?? POW (concr o1) (concr o2) (concr o3) f1\sub\c f2\sub\c);]     
 qed.
 
 theorem BP_to_OBP_faithful:
  ∀S,T.∀f,g:arrows2 (category2_of_category1 BP) S T.
   map_arrows2 ?? BP_to_OBP ?? f = map_arrows2 ?? BP_to_OBP ?? g → f=g.
  intros; change with ( (⊩) ∘ f \sub \c = (⊩) ∘ g \sub \c);
- apply (SUBSETS_faithful);
- apply (.= respects_comp2 ?? SUBSETS' (concr S) (concr T) (form T) f \sub \c (⊩ \sub T));
+ apply (POW_faithful);
+ apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) f \sub \c (⊩ \sub T));
  apply sym2;
- apply (.= respects_comp2 ?? SUBSETS' (concr S) (concr T) (form T) g \sub \c (⊩ \sub T));
+ apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) g \sub \c (⊩ \sub T));
  apply sym2;
  apply e;
 qed.
 
 theorem BP_to_OBP_full: ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BP_to_OBP S T g = f).
  intros; 
- cases (SUBSETS_full (concr S) (concr T) (Oconcr_rel ?? f)) (gc Hgc);
- cases (SUBSETS_full (form S) (form T) (Oform_rel ?? f)) (gf Hgf);
+ cases (POW_full (concr S) (concr T) (Oconcr_rel ?? f)) (gc Hgc);
+ cases (POW_full (form S) (form T) (Oform_rel ?? f)) (gf Hgf);
  exists[
    constructor 1; [apply gc|apply gf]
-   apply (SUBSETS_faithful);
-   apply (let xxxx ≝SUBSETS' in .= respects_comp2 ?? SUBSETS' (concr S) (concr T) (form T) gc (rel T));
+   apply (POW_faithful);
+   apply (let xxxx ≝POW in .= respects_comp2 ?? POW (concr S) (concr T) (form T) gc (rel T));
    apply rule (.= Hgc‡#);
    apply (.= Ocommute ?? f);
    apply (.= #‡Hgf^-1);
-   apply (let xxxx ≝SUBSETS' in (respects_comp2 ?? SUBSETS' (concr S) (form S) (form T) (rel S) gf)^-1)]
+   apply (let xxxx ≝POW in (respects_comp2 ?? POW (concr S) (form S) (form T) (rel S) gf)^-1)]
  split;
   [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x); 
   | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
diff --git a/helm/software/matita/contribs/formal_topology/overlap/categories.ma b/helm/software/matita/contribs/formal_topology/overlap/categories.ma
index af58968fc..17c5f498f 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/categories.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/categories.ma
@@ -100,6 +100,31 @@ 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 }.
+
+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 =_0 b)" non associative with precedence 45
+for @{ eq_rel ? (eq ?) $a $b }.
+notation > "hvbox(a break =_1 b)" non associative with precedence 45
+for @{ eq_rel1 ? (eq1 ?) $a $b }.
+notation > "hvbox(a break =_2 b)" non associative with precedence 45
+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).
diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
index b278adfc0..99ea22e91 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
@@ -65,8 +65,8 @@ record OAlgebra : Type2 := {
   oa_overlap_sym: ∀a,b:oa_P.oa_overlap a b → oa_overlap b a;
   oa_meet_inf: 
     ∀I:SET.∀p_i:I ⇒ oa_P.∀p:oa_P.
-      oa_leq p (oa_meet I p_i) = ∀i:I.oa_leq p (p_i i);
-  oa_join_sup: ∀I:SET.∀p_i:I ⇒ oa_P.∀p:oa_P.oa_leq (oa_join I p_i) p = ∀i:I.oa_leq (p_i i) p;
+      oa_leq p (oa_meet I p_i) = (∀i:I.oa_leq p (p_i i));
+  oa_join_sup: ∀I:SET.∀p_i:I ⇒ oa_P.∀p:oa_P.oa_leq (oa_join I p_i) p = (∀i:I.oa_leq (p_i i) p);
   oa_zero_bot: ∀p:oa_P.oa_leq oa_zero p;
   oa_one_top: ∀p:oa_P.oa_leq p oa_one;
   oa_overlap_preserves_meet_: 
@@ -74,7 +74,7 @@ record OAlgebra : Type2 := {
        (oa_meet ? { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p oa_P p q });
   oa_join_split:
       ∀I:SET.∀p.∀q:I ⇒ oa_P.
-       oa_overlap p (oa_join I q) = ∃i:I.oa_overlap p (q i);
+       oa_overlap p (oa_join I q) = (∃i:I.oa_overlap p (q i));
   (*oa_base : setoid;
   1) enum non e' il nome giusto perche' non e' suriettiva
   2) manca (vedere altro capitolo) la "suriettivita'" come immagine di insiemi di oa_base
@@ -187,10 +187,11 @@ constructor 1;
 | constructor 1;
    (* tenere solo una uguaglianza e usare la proposizione 9.9 per
       le altre (unicita' degli aggiunti e del simmetrico) *)
-   [ apply (λp,q. And42 (eq2 ? (or_f_minus_star_ ?? p) (or_f_minus_star_ ?? q)) 
-             (eq2 ? (or_f_minus_ ?? p) (or_f_minus_ ?? q)) 
-             (eq2 ? (or_f_ ?? p) (or_f_ ?? q)) 
-             (eq2 ? (or_f_star_ ?? p) (or_f_star_ ?? q))); 
+   [ apply (λp,q. And42 
+             (or_f_minus_star_ ?? p = or_f_minus_star_ ?? q) 
+             (or_f_minus_ ?? p = or_f_minus_ ?? q) 
+             (or_f_ ?? p = or_f_ ?? q) 
+             (or_f_star_ ?? p = or_f_star_ ?? q)); 
    | whd; simplify; intros; repeat split; intros; apply refl2;
    | whd; simplify; intros; cases a; clear a; split; 
      intro a; apply sym1; generalize in match a;assumption;
diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma
index b6d6e1b3d..58725373c 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma
@@ -22,8 +22,8 @@ record Obasic_pair: Type2 ≝
  }.
 
 (* FIX *)
-interpretation "basic pair relation indexed" 'Vdash2 x y c = (Orel c x y).
-interpretation "basic pair relation (non applied)" 'Vdash c = (Orel c).
+interpretation "o-basic pair relation indexed" 'Vdash2 x y c = (Orel c x y).
+interpretation "o-basic pair relation (non applied)" 'Vdash c = (Orel c).
 
 alias symbol "eq" = "setoid1 eq".
 alias symbol "compose" = "category1 composition".
@@ -38,8 +38,8 @@ record Orelation_pair (BP1,BP2: Obasic_pair): Type2 ≝
  }.
  
 (* FIX *)
-interpretation "concrete relation" 'concr_rel r = (Oconcr_rel __ r). 
-interpretation "formal relation" 'form_rel r = (Oform_rel __ r). 
+interpretation "o-concrete relation" 'concr_rel r = (Oconcr_rel __ r). 
+interpretation "o-formal relation" 'form_rel r = (Oform_rel __ r). 
 
 definition Orelation_pair_equality:
  ∀o1,o2. equivalence_relation2 (Orelation_pair o1 o2).
diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma b/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
index b3939f90b..af3015eb3 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
@@ -15,7 +15,7 @@
 include "relations.ma".
 include "o-algebra.ma".
 
-definition SUBSETS: objs1 SET → OAlgebra.
+definition POW': objs1 SET → OAlgebra.
  intro A; constructor 1;
   [ apply (Ω \sup A);
   | apply subseteq;
@@ -42,16 +42,16 @@ definition SUBSETS: objs1 SET → OAlgebra.
   | intros; split; intro;
      [ cases f; cases x1; exists [apply w1] exists [apply w] assumption;
      | cases e; cases x; exists; [apply w1] [ assumption | exists; [apply w] assumption]]
-  | intros; intros 2; cases (f (singleton ? a) ?);
+  | intros; intros 2; cases (f {(a)} ?); 
      [ exists; [apply a] [assumption | change with (a = a); apply refl1;]
      | change in x1 with (a = w); change with (mem A a q); apply (. (x1‡#));
        assumption]]
 qed.
 
-definition powerset_of_SUBSETS: ∀A.oa_P (SUBSETS A) → Ω \sup A ≝ λA,x.x.
-coercion powerset_of_SUBSETS.
+definition powerset_of_POW': ∀A.oa_P (POW' A) → Ω \sup A ≝ λA,x.x.
+coercion powerset_of_POW'.
 
-definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → arrows2 OA (SUBSETS o1) (SUBSETS o2).
+definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → arrows2 OA (POW' o1) (POW' o2).
  intros;
  constructor 1;
   [ constructor 1; 
@@ -112,7 +112,7 @@ lemma orelation_of_relation_preserves_equality:
 qed.
 
 lemma orelation_of_relation_preserves_identity:
- ∀o1:REL. eq2 ? (orelation_of_relation ?? (id1 ? o1)) (id2 OA (SUBSETS o1)).
+ ∀o1:REL. eq2 ? (orelation_of_relation ?? (id1 ? o1)) (id2 OA (POW' o1)).
  intros; split; intro; 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;
@@ -141,7 +141,7 @@ alias symbol "compose" = "category1 composition".
 lemma orelation_of_relation_preserves_composition:
  ∀o1,o2,o3:REL.∀F: arrows1 ? o1 o2.∀G: arrows1 ? o2 o3.
   orelation_of_relation ?? (G ∘ F) =
-   comp2 OA (SUBSETS o1) (SUBSETS o2) (SUBSETS o3)
+   comp2 OA (POW' o1) (POW' o2) (POW' o3)
     ?? (*(orelation_of_relation ?? F) (orelation_of_relation ?? G)*).
  [ apply (orelation_of_relation ?? F); | apply (orelation_of_relation ?? G); ]
  intros; split; intro; split; whd; intro; whd in ⊢ (% → %); intros;
@@ -161,9 +161,9 @@ lemma orelation_of_relation_preserves_composition:
   | cases f1; clear f1; cases x; clear x; apply (f w); assumption;]
 qed.
 
-definition SUBSETS': carr3 (arrows3 CAT2 (category2_of_category1 REL) OA).
+definition POW: carr3 (arrows3 CAT2 (category2_of_category1 REL) OA).
  constructor 1;
-  [ apply SUBSETS;
+  [ apply POW';
   | intros; constructor 1;
      [ apply (orelation_of_relation S T);
      | intros; apply (orelation_of_relation_preserves_equality S T a a' e); ]
@@ -171,22 +171,24 @@ definition SUBSETS': carr3 (arrows3 CAT2 (category2_of_category1 REL) OA).
   | apply orelation_of_relation_preserves_composition; ]
 qed.
 
-theorem SUBSETS_faithful:
+theorem POW_faithful:
  ∀S,T.∀f,g:arrows2 (category2_of_category1 REL) S T.
-  map_arrows2 ?? SUBSETS' ?? f = map_arrows2 ?? SUBSETS' ?? g → f=g.
- intros; unfold SUBSETS' in e; simplify in e; cases e;
+  map_arrows2 ?? POW ?? f = map_arrows2 ?? POW ?? g → f=g.
+ intros; 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; lapply (e3 (singleton ? x)); cases Hletin;
+ intros 2; 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 Hletin1; cases x1; change in f3 with (eq1 ? x w); apply (. f3‡#); assumption;]
+  |*: cases Hletin; cases x1; change in f3 with (x =_1 w); apply (. f3‡#); assumption;]
 qed.
 
+interpretation "lifting singl" 'singl x = 
+ (fun11 _ (objs2 (POW _))  (singleton _) x).
 
-theorem SUBSETS_full: ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? SUBSETS' S T g = f).
+theorem POW_full: ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? POW S T g = f).
  intros; exists;
   [ constructor 1; constructor 1;
-     [ apply (λx:carr S.λy:carr T. y ∈ f (singleton S x));
+     [ apply (λx:carr S.λy:carr T. y ∈ f {(x)});
      | 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 ]]  
diff --git a/helm/software/matita/contribs/formal_topology/overlap/subsets.ma b/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
index e6d187216..4152f4852 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
@@ -129,7 +129,7 @@ interpretation "union" 'union U V = (fun21 ___ (union _) U V).
 (* qua non riesco a mettere set *)
 definition singleton: ∀A:setoid. unary_morphism1 A (Ω \sup A).
  intros; constructor 1;
-  [ apply (λa:A.{b | eq ? a b}); unfold setoid1_of_setoid; simplify;
+  [ apply (λa:A.{b | a =_0 b}); unfold setoid1_of_setoid; simplify;
     intros; simplify;
     split; intro;
     apply (.= e1);