X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fsetoids1.ma;h=068334183dc698ebb9cb3dea09ed1e7942a87c28;hb=2c486bbea1d6ffb072d0ff83f9df129b7860f3e1;hp=bae7f549ecc5ca3fdf800b59788aa5b3eaf48399;hpb=6678a28314d8878bb46d5de7b1060628f4930242;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/setoids1.ma b/helm/software/matita/nlibrary/sets/setoids1.ma
index bae7f549e..068334183 100644
--- a/helm/software/matita/nlibrary/sets/setoids1.ma
+++ b/helm/software/matita/nlibrary/sets/setoids1.ma
@@ -14,52 +14,120 @@
 
 include "properties/relations1.ma".
 include "sets/setoids.ma".
+include "hints_declaration.ma".
 
-nrecord setoid1: Type[2] ≝
- { carr1:> Type[1];
-   eq1: equivalence_relation1 carr1
- }.
+nrecord setoid1: Type[2] ≝ { 
+  carr1:> Type[1];
+  eq1: equivalence_relation1 carr1 
+}.
 
+unification hint 0 ≔ R : setoid1; 
+   MR ≟ (carr1 R), 
+   lock ≟ mk_lock2 Type[1] MR setoid1 R 
+(* ---------------------------------------- *) ⊢ 
+   setoid1 ≡ force2 ? MR lock.
+
+notation < "[\setoid1\ensp\of term 19 x]" non associative with precedence 90 for @{'mk_setoid1 $x}.
+interpretation "mk_setoid1" 'mk_setoid1 x = (mk_setoid1 x ?).
+
+(* da capire se mettere come coercion *)
 ndefinition setoid1_of_setoid: setoid → setoid1.
- #s; napply mk_setoid1
-  [ napply (carr s)
-  | napply (mk_equivalence_relation1 s)
-    [ napply eq
-    | napply refl
-    | napply sym
-    | napply trans]##]
+ #s; @ (carr s); @ (eq0…) (refl…) (sym…) (trans…);
 nqed.
 
-ncoercion setoid1_of_setoid : ∀s:setoid. setoid1 ≝ setoid1_of_setoid
- on _s: setoid to setoid1.
-(*prefer coercion Type_OF_setoid.*)
+alias symbol "hint_decl" = "hint_decl_CProp2".
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type2".
+unification hint 0 ≔ A,x,y;
+   T  ≟ carr A, 
+   R  ≟ setoid1_of_setoid A,
+   T1 ≟ carr1 R
+(*-----------------------------------------------*) ⊢
+   eq_rel T (eq0 A) x y ≡ eq_rel1 T1 (eq1 R) x y.
+
+unification hint 0 ≔ A;
+   R  ≟ setoid1_of_setoid A
+(*-----------------------------------------------*) ⊢
+   carr A ≡ carr1 R.
 
 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).
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq0 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 }.
 notation > "hvbox(a break =_0 b)" non associative with precedence 45
-for @{ eq_rel ? (eq ?) $a $b }.
+for @{ eq_rel ? (eq0 ?) $a $b }.
 notation > "hvbox(a break =_1 b)" non associative with precedence 45
 for @{ eq_rel1 ? (eq1 ?) $a $b }.
 
 interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
 interpretation "setoid symmetry" 'invert r = (sym ???? r).
-notation ".= r" with precedence 50 for @{'trans $r}.
+notation ".=_1 r" with precedence 50 for @{'trans_x1 $r}.
 interpretation "trans1" 'trans r = (trans1 ????? r).
 interpretation "trans" 'trans r = (trans ????? r).
+interpretation "trans1_x1" 'trans_x1 r = (trans1 ????? r).
 
-nrecord unary_morphism1 (A,B: setoid1) : Type[1] ≝
- { fun11:1> A → B;
-   prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
- }.
-
-nrecord binary_morphism1 (A,B,C:setoid1) : Type[1] ≝
- { fun21:2> A → B → C;
-   prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
- }.
-
+nrecord unary_morphism1 (A,B: setoid1) : Type[1] ≝ { 
+  fun11:1> A → B;
+  prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a') 
+}.
+ 
+notation > "B ⇒_1 C" right associative with precedence 72 for @{'umorph1 $B $C}.
+notation "hvbox(B break ⇒\sub 1 C)" right associative with precedence 72 for @{'umorph1 $B $C}.
+interpretation "unary morphism 1" 'umorph1 A B = (unary_morphism1 A B).
+ 
+notation "┼_1 c" with precedence 89 for @{'prop1_x1 $c }.
 interpretation "prop11" 'prop1 c = (prop11 ????? c).
-interpretation "prop21" 'prop2 l r = (prop21 ???????? l r).
+interpretation "prop11_x1" 'prop1_x1 c = (prop11 ????? c).
 interpretation "refl1" 'refl = (refl1 ???).
+
+ndefinition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
+ #s; #s1; @ (s ⇒_1 s1); @
+     [ #f; #g; napply (∀a,a':s. a=a' → f a = g a')
+     | #x; #a; #a'; #Ha; napply (.= †Ha); napply refl1
+     | #x; #y; #H; #a; #a'; #Ha; napply (.= †Ha); napply sym1; /2/
+     | #x; #y; #z; #H1; #H2; #a; #a'; #Ha; napply (.= †Ha); napply trans1; ##[##2: napply H1 | ##skip | napply H2]//;##]
+nqed.
+
+unification hint 0 ≔ S, T ;
+   R ≟ (unary_morphism1_setoid1 S T)
+(* --------------------------------- *) ⊢
+   carr1 R ≡ S ⇒_1 T.
+   
+notation "l ╪_1 r" with precedence 89 for @{'prop2_x1 $l $r }.
+interpretation "prop21" 'prop2 l r = (prop11 ? (unary_morphism1_setoid1 ??) ? ?? l ?? r).
+interpretation "prop21_x1" 'prop2_x1 l r = (prop11 ? (unary_morphism1_setoid1 ??) ? ?? l ?? r).
+
+nlemma unary_morph1_eq1: ∀A,B.∀f,g: A ⇒_1 B. (∀x. f x = g x) → f = g.
+/3/. nqed.
+
+nlemma mk_binary_morphism1:
+ ∀A,B,C: setoid1. ∀f: A → B → C. (∀a,a',b,b'. a=a' → b=b' → f a b = f a' b') →
+  A ⇒_1 (unary_morphism1_setoid1 B C).
+ #A; #B; #C; #f; #H; @ [ #x; @ (f x) ] #a; #a'; #Ha [##2: napply unary_morph1_eq1; #y]
+ /2/.
+nqed.
+
+ndefinition composition1 ≝
+ λo1,o2,o3:Type[1].λf:o2 → o3.λg: o1 → o2.λx.f (g x).
+ 
+interpretation "function composition" 'compose f g = (composition ??? f g).
+interpretation "function composition1" 'compose f g = (composition1 ??? f g).
+
+ndefinition comp1_unary_morphisms: 
+  ∀o1,o2,o3:setoid1.o2 ⇒_1 o3 → o1 ⇒_1 o2 → o1 ⇒_1 o3.
+#o1; #o2; #o3; #f; #g; @ (f ∘ g);
+ #a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
+nqed.
+
+unification hint 0 ≔ o1,o2,o3:setoid1,f:o2 ⇒_1 o3,g:o1 ⇒_1 o2;
+   R ≟ (mk_unary_morphism1 ?? (composition1 ??? f g)
+        (prop11 ?? (comp1_unary_morphisms o1 o2 o3 f g)))
+ (* -------------------------------------------------------------------- *) ⊢
+                              fun11 ?? R ≡ (composition1 ??? f g).
+                              
+ndefinition comp1_binary_morphisms:
+ ∀o1,o2,o3. (o2 ⇒_1 o3) ⇒_1 ((o1 ⇒_1 o2) ⇒_1 (o1 ⇒_1 o3)).
+#o1; #o2; #o3; napply mk_binary_morphism1
+ [ #f; #g; napply (comp1_unary_morphisms … f g) 
+ | #a; #a'; #b; #b'; #ea; #eb; #x; #x'; #Hx; nnormalize; /3/ ]
+nqed.