o-basic_pairs are indeed examples of o-basic_topologies!

[helm.git] / helm / software / matita / contribs / formal_topology / overlap / categories.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/categories.ma b/helm/software/matita/contribs/formal_topology/overlap/categories.ma
index 75ca005c64b9ce03d24b487c22d103ee97499492..e22402d9f52542745b29e226597fe74a14920622 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/categories.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/categories.ma
@@ -54,8 +54,8 @@ definition setoid1_of_setoid: setoid → setoid1.
     | apply (trans s)]]
 qed.
 
-(* questa coercion e' necessaria per problemi di unificazione *)
 coercion setoid1_of_setoid.
+prefer coercion Type_OF_setoid.
 
 definition reflexive2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λA:Type2.λR:A→A→CProp2.∀x:A.R x x.
 definition symmetric2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λC:Type2.λlt:C→C→CProp2. ∀x,y:C.lt x y → lt y x.
@@ -85,25 +85,11 @@ definition setoid2_of_setoid1: setoid1 → setoid2.
     | apply (trans1 s)]]
 qed.
 
-(*coercion setoid2_of_setoid1.*)
-
-(*
-definition Leibniz: Type → setoid.
- intro;
- constructor 1;
-  [ apply T
-  | constructor 1;
-     [ apply (λx,y:T.cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y)
-     | alias id "refl_eq" = "cic:/matita/logic/equality/eq.ind#xpointer(1/1/1)".
-       apply refl_eq
-     | alias id "sym_eq" = "cic:/matita/logic/equality/sym_eq.con".
-       apply sym_eq
-     | alias id "trans_eq" = "cic:/matita/logic/equality/trans_eq.con".
-       apply trans_eq ]]
-qed.
-
-coercion Leibniz.
-*)
+coercion setoid2_of_setoid1.
+prefer coercion Type_OF_setoid2. 
+prefer coercion Type_OF_setoid. 
+prefer coercion Type_OF_setoid1. 
+(* we prefer 0 < 1 < 2 *)
 
 interpretation "setoid2 eq" 'eq x y = (eq_rel2 _ (eq2 _) x y).
 interpretation "setoid1 eq" 'eq x y = (eq_rel1 _ (eq1 _) x y).
@@ -171,12 +157,12 @@ definition CPROP: setoid1.
 qed.
 
 alias symbol "eq" = "setoid1 eq".
-definition if': ∀A,B:CPROP. A = B → A → B.
- intros; apply (if ?? e); assumption.
+definition fi': ∀A,B:CPROP. A = B → B → A.
+ intros; apply (fi ?? e); assumption.
 qed.
 
-notation ". r" with precedence 50 for @{'if $r}.
-interpretation "if" 'if r = (if' __ r).
+notation ". r" with precedence 50 for @{'fi $r}.
+interpretation "fi" 'fi r = (fi' __ r).
 
 definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
  constructor 1;
@@ -210,21 +196,6 @@ definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
      | apply (fi ?? e1); apply f; apply (if ?? e); assumption]]
 qed.
 
-(*
-definition eq_morphism: ∀S:setoid. binary_morphism S S CPROP.
- intro;
- constructor 1;
-  [ apply (eq_rel ? (eq S))
-  | intros; split; intro;
-     [ apply (.= H \sup -1);
-       apply (.= H2);
-       assumption
-     | apply (.= H);
-       apply (.= H2);
-       apply (H1 \sup -1)]]
-qed.
-*)
-
 record category : Type1 ≝
  { objs:> Type0;
    arrows: objs → objs → setoid;
@@ -267,9 +238,7 @@ interpretation "category1 assoc" 'assoc = (comp_assoc1 ________).
 interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x).
 interpretation "category assoc" 'assoc = (comp_assoc ________).
 
-(* bug grande come una casa?
-   Ma come fa a passare la quantificazione larga??? *)
-definition unary_morphism_setoid: setoid → setoid → setoid1.
+definition unary_morphism_setoid: setoid → setoid → setoid.
  intros;
  constructor 1;
   [ apply (unary_morphism s s1);
@@ -300,25 +269,10 @@ definition setoid_of_SET: objs1 SET → setoid.
  intros; apply o; qed.
 coercion setoid_of_SET.
 
-definition setoid1_of_SET: SET → setoid1.
- intro; whd in t; apply setoid1_of_setoid; apply t.
-qed.
-coercion setoid1_of_SET.
-
-definition eq': ∀w:SET.equivalence_relation ? := λw.eq w.
-
-definition prop1_SET : 
- ∀A,B:SET.∀w:arrows1 SET A B.∀a,b:Type_OF_objs1 A.eq' ? a b→eq' ? (w a) (w b).
-intros; apply (prop1 A B w a b e);
-qed.
-
-
-interpretation "SET dagger" 'prop1 h = (prop1_SET _ _ _ _ _ h).
 notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
 interpretation "unary morphism" 'Imply a b = (arrows1 SET a b).
-interpretation "SET eq" 'eq x y = (eq_rel _ (eq' _) x y).
 
-definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid2.
+definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
  intros;
  constructor 1;
   [ apply (unary_morphism1 s s1);
@@ -351,14 +305,7 @@ definition setoid1_OF_SET1: objs2 SET1 → setoid1.
  intros; apply o; qed.
 
 coercion setoid1_OF_SET1.
-
-definition eq'': ∀w:SET1.equivalence_relation1 ? := λw.eq1 w.
-
-definition prop11_SET1 : 
- ∀A,B:SET1.∀w:arrows2 SET1 A B.∀a,b:Type_OF_objs2 A.eq'' ? a b→eq'' ? (w a) (w b).
-intros; apply (prop11 A B w a b e);
-qed.
-
+ 
 definition setoid2_OF_category2: Type_OF_category2 SET1 → setoid2.
  intro; apply (setoid2_of_setoid1 t); qed.
 coercion setoid2_OF_category2.
@@ -373,14 +320,15 @@ qed.
 coercion Type1_OF_SET1.
 
 definition Type_OF_setoid1_of_carr: ∀U. carr U → Type_OF_setoid1 ?(*(setoid1_of_SET U)*).
- [ apply setoid1_of_SET; apply U
+ [ apply rule U; 
  | intros; apply c;]
 qed.
 coercion Type_OF_setoid1_of_carr.
 
-interpretation "SET dagger" 'prop1 h = (prop11_SET1 _ _ _ _ _ h).
+definition carr' ≝ λx:Type_OF_category1 SET.Type_OF_Type0 (carr x).
+coercion carr'. (* we prefer the lower carrier projection *)
+
 interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
-interpretation "SET1 eq" 'eq x y = (eq_rel1 _ (eq'' _) x y).
 
 lemma unary_morphism1_of_arrows1_SET1: ∀S,T. (S ⇒ T) → unary_morphism1 S T.
  intros; apply t;