Setoids, setoids1, sets, and the like. The mess begins.

[helm.git] / helm / software / matita / nlibrary / sets / setoids.ma

diff --git a/helm/software/matita/nlibrary/sets/setoids.ma b/helm/software/matita/nlibrary/sets/setoids.ma
index 9d31b09439dc29f0b12eb9550140f9918f41ac3a..28761806f0a9bab4d7f4e34eb0f858678d5c3db1 100644 (file)
--- a/helm/software/matita/nlibrary/sets/setoids.ma
+++ b/helm/software/matita/nlibrary/sets/setoids.ma
@@ -15,63 +15,43 @@
 include "logic/connectives.ma".
 include "properties/relations.ma".
 
-nrecord setoid : Type[1] ≝
- { carr:> Type;
-   eq: carr → carr → CProp;
-   refl: reflexive ? eq;
-   sym: symmetric ? eq;
-   trans: transitive ? eq
- }.
+(*
+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 }.
 
-ndefinition proofs: CProp → setoid.
-#P; napply (mk_setoid ?????);
-##[ napply P;
-##| #x; #y; napply True;
-##|##*: nwhd; nrepeat (#_); napply I;
-##]
-nqed.
+interpretation "ID eq" 'eqID x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y).
+*)
 
-nrecord function_space (A,B: setoid): Type ≝
- { f:1> A → B;
-   f_ok: ∀a,a':A. proofs (eq ? a a') → proofs (eq ? (f a) (f a'))
+nrecord setoid : Type[1] ≝
+ { carr:> Type[0];
+   eq: equivalence_relation carr
  }.
 
-notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
+
+notation > "hvbox(a break =_0 b)" non associative with precedence 45
+for @{ eq_rel ? (eq ?) $a $b }.
 
-ndefinition function_space_setoid: setoid → setoid → setoid.
- #A; #B; napply (mk_setoid ?????);
-##[ napply (function_space A B);
-##| #f; #f1; napply (∀a:A. proofs (eq ? (f a) (f1 a)));
-##| nwhd; #x; #a;
-    napply (f_ok ? ? x ? ? ?); (* QUI!! *)
-(*    unfold carr; unfold proofs; simplify;
-    apply (refl A)
-  | simplify;
-    intros;
-    unfold carr; unfold proofs; simplify;
-    apply (sym B);
-    apply (f a)
-  | simplify;
-    intros;
-    unfold carr; unfold proofs; simplify;
-    apply (trans B ? (y a));
-    [ apply (f a)
-    | apply (f1 a)]]
-qed.
+interpretation "setoid symmetry" 'invert r = (sym ???? r).
+notation ".= r" with precedence 50 for @{'trans $r}.
+interpretation "trans" 'trans r = (trans ????? r).
 
-nrecord isomorphism (A,B: setoid): Type ≝
- { map1:> function_space_setoid A B;
-   map2:> function_space_setoid B A;
-   inv1: ∀a:A. proofs (eq ? (map2 (map1 a)) a);
-   inv2: ∀b:B. proofs (eq ? (map1 (map2 b)) b)
+nrecord unary_morphism (A,B: setoid) : Type[0] ≝
+ { fun1:1> A → B;
+   prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
  }.
 
-interpretation "isomorphism" 'iff x y = (isomorphism x y).
+nrecord binary_morphism (A,B,C:setoid) : Type[0] ≝
+ { fun2:2> A → B → C;
+   prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
+ }.
 
-(*
-record dependent_product (A:setoid)  (B: A ⇒ setoids): Type ≝
- { dp:> ∀a:A.carr (B a);
-   dp_ok: ∀a,a':A. ∀p:proofs1 (eq1 ? a a'). proofs1 (eq1 ? (dp a) (map2 ?? (f1_ok ?? B ?? p) (dp a')))
- }.*)
- 
- *)
\ No newline at end of file
+notation "† c" with precedence 90 for @{'prop1 $c }.
+notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
+notation "#" with precedence 90 for @{'refl}.
+interpretation "prop1" 'prop1 c  = (prop1 ????? c).
+interpretation "prop2" 'prop2 l r = (prop2 ???????? l r).
+interpretation "refl" 'refl = (refl ???).