X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fsetoids.ma;h=e40dad6f6d20e4e3a1f7624b6f28b5d6056a6ef3;hb=1d7773584ddd6463b0941026f114b0173e3b6b72;hp=ed2973d87af9170b0a910a747f42f64f343e8ce6;hpb=1dd64d6c49db7dc0dc0ee39c30da4c7a043b8bde;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/setoids.ma b/helm/software/matita/nlibrary/sets/setoids.ma
index ed2973d87..e40dad6f6 100644
--- a/helm/software/matita/nlibrary/sets/setoids.ma
+++ b/helm/software/matita/nlibrary/sets/setoids.ma
@@ -14,64 +14,108 @@
 
 include "logic/connectives.ma".
 include "properties/relations.ma".
+include "hints_declaration.ma".
 
-nrecord setoid : Type[1] ≝
- { carr:> Type;
-   eq: carr → carr → CProp;
-   refl: reflexive ? eq;
-   sym: symmetric ? eq;
-   trans: transitive ? eq
- }.
-
-ndefinition proofs: CProp → setoid.
-#P; napply (mk_setoid ?????);
-##[ napply P;
-##| #x; #y; napply True;
-##|##*: nwhd; nrepeat (#_); napply I;
-##]
+nrecord setoid : Type[1] ≝ { 
+  carr:> Type[0];
+  eq0: equivalence_relation carr
+}.
+
+(* activate non uniform coercions on: Type → setoid *)
+unification hint 0 ≔ R : setoid;
+   MR ≟ carr R, 
+   lock ≟ mk_lock1 Type[0] MR setoid R 
+(* ---------------------------------------- *) ⊢ 
+   setoid ≡ force1 ? MR lock.
+
+notation < "[\setoid\ensp\of term 19 x]" non associative with precedence 90 for @{'mk_setoid $x}.
+interpretation "mk_setoid" 'mk_setoid x = (mk_setoid x ?).
+
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq0 t) x y).
+(* single = is for the abstract equality of setoids, == is for concrete 
+   equalities (that may be lifted to the setoid level when needed *)
+notation < "hvbox(a break mpadded width -50% (=)= b)" non associative with precedence 45 for @{ 'eq_low $a $b }.
+notation > "a == b" non associative with precedence 45 for @{ 'eq_low $a $b }.
+
+notation > "hvbox(a break =_0 b)" non associative with precedence 45
+for @{ eq_rel ? (eq0 ?) $a $b }.
+
+interpretation "setoid symmetry" 'invert r = (sym ???? r).
+notation ".= r" with precedence 50 for @{'trans $r}.
+interpretation "trans" 'trans r = (trans ????? r).
+notation > ".=_0 r" with precedence 50 for @{'trans_x0 $r}.
+interpretation "trans_x0" 'trans_x0 r = (trans ????? r).
+
+nrecord unary_morphism (A,B: setoid) : Type[0] ≝ { 
+  fun1:1> A → B;
+  prop1: ∀a,a'. a = a' → fun1 a = fun1 a'
+}.
+
+notation > "B ⇒_0 C" right associative with precedence 72 for @{'umorph0 $B $C}.
+notation "hvbox(B break ⇒\sub 0 C)" right associative with precedence 72 for @{'umorph0 $B $C}.
+interpretation "unary morphism 0" 'umorph0 A B = (unary_morphism A B).
+
+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 "refl" 'refl = (refl ???).
+notation "┼_0 c" with precedence 89 for @{'prop1_x0 $c }.
+notation "l ╪_0 r" with precedence 89 for @{'prop2_x0 $l $r }.
+interpretation "prop1_x0" 'prop1_x0 c  = (prop1 ????? c).
+
+ndefinition unary_morph_setoid : setoid → setoid → setoid.
+#S1; #S2; @ (S1 ⇒_0 S2); @;
+##[ #f; #g; napply (∀x,x'. x=x' → f x = g x');
+##| #f; #x; #x'; #Hx; napply (.= †Hx); napply #;
+##| #f; #g; #H; #x; #x'; #Hx; napply ((H … Hx^-1)^-1);
+##| #f; #g; #h; #H1; #H2; #x; #x'; #Hx; napply (trans … (H1 …) (H2 …)); //; ##]
 nqed.
 
-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'))
- }.
-
-notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $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.
-
-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)
- }.
-
-interpretation "isomorphism" 'iff x y = (isomorphism x y).
-
-(*
-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')))
- }.*)
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
+unification hint 0 ≔ o1,o2 ; 
+     X ≟ unary_morph_setoid o1 o2
+  (* ----------------------------- *) ⊢ 
+     carr X ≡ o1 ⇒_0 o2.
+
+interpretation "prop2" 'prop2 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
+interpretation "prop2_x0" 'prop2_x0 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
+
+nlemma unary_morph_eq: ∀A,B.∀f,g:A ⇒_0 B. (∀x. f x = g x) → f = g.
+#A B f g H x1 x2 E; napply (.= †E); napply H; nqed.
+
+nlemma mk_binary_morphism:
+ ∀A,B,C: setoid. ∀f: A → B → C. (∀a,a',b,b'. a=a' → b=b' → f a b = f a' b') →
+  A ⇒_0 (unary_morph_setoid B C).
+ #A; #B; #C; #f; #H; @; ##[ #x; @ (f x) ] #a; #a'; #Ha [##2: napply unary_morph_eq; #y]
+ /2/.
+nqed.
+
+ndefinition composition ≝
+ λo1,o2,o3:Type[0].λf:o2 → o3.λg: o1 → o2.λx.f (g x).
  
- *)
\ No newline at end of file
+interpretation "function composition" 'compose f g = (composition ??? f g).
+
+ndefinition comp_unary_morphisms:
+ ∀o1,o2,o3:setoid.o2 ⇒_0 o3 → o1 ⇒_0 o2 → o1 ⇒_0  o3.
+#o1; #o2; #o3; #f; #g; @ (f ∘ g);
+ #a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
+nqed.
+
+unification hint 0 ≔ o1,o2,o3:setoid,f:o2 ⇒_0 o3,g:o1 ⇒_0 o2;
+   R ≟ mk_unary_morphism o1 o3 
+        (composition ??? (fun1 o2 o3 f) (fun1 o1 o2 g))
+        (prop1 o1 o3 (comp_unary_morphisms o1 o2 o3 f g))
+ (* -------------------------------------------------------------------- *) ⊢
+    fun1 o1 o3 R ≡ composition ??? (fun1 o2 o3 f) (fun1 o1 o2 g).
+
+ndefinition comp_binary_morphisms: 
+  ∀o1,o2,o3.(o2 ⇒_0 o3) ⇒_0 ((o1 ⇒_0 o2) ⇒_0 (o1 ⇒_0 o3)).
+#o1; #o2; #o3; napply mk_binary_morphism
+ [ #f; #g; napply (comp_unary_morphisms ??? f g) 
+         (* CSC: why not ∘? 
+            GARES: because the coercion to FunClass is not triggered if there
+                   are no "extra" arguments. We could fix that in the refiner
+         *)
+ | #a; #a'; #b; #b'; #ea; #eb; #x; #x'; #Hx; nnormalize; /3/ ]
+nqed.