X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fsetoids1.ma;h=90be6bc94be18f5758c7ce8b8fb5ee8c2d63c5a4;hb=e880d6eab5e1700f4a625ddcd7d0fa8f0cce2dcc;hp=0571b6dfa8e5e2c890657ff682fc14f5341e5592;hpb=1dd64d6c49db7dc0dc0ee39c30da4c7a043b8bde;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/setoids1.ma b/helm/software/matita/nlibrary/sets/setoids1.ma
index 0571b6dfa..90be6bc94 100644
--- a/helm/software/matita/nlibrary/sets/setoids1.ma
+++ b/helm/software/matita/nlibrary/sets/setoids1.ma
@@ -12,165 +12,122 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "properties/relations1.ma".
 include "sets/setoids.ma".
-
-(*
-definition reflexive1 ≝ λA:Type.λR:A→A→CProp.∀x:A.R x x.
-definition symmetric1 ≝ λC:Type.λlt:C→C→CProp. ∀x,y:C.lt x y → lt y x.
-definition transitive1 ≝ λA:Type.λR:A→A→CProp.∀x,y,z:A.R x y → R y z → R x z.
-
-record setoid1 : Type ≝
- { carr1:> Type;
-   eq1: carr1 → carr1 → CProp;
-   refl1: reflexive1 ? eq1;
-   sym1: symmetric1 ? eq1;
-   trans1: transitive1 ? eq1
- }.
-
-definition proofs1: CProp → setoid1.
- intro;
- constructor 1;
-  [ apply A
-  | intros;
-    apply True
-  | intro;
-    constructor 1
-  | intros 3;
-    constructor 1
-  | intros 5;
-    constructor 1]
-qed.
-
-ndefinition CCProp: setoid1.
- constructor 1;
-  [ apply CProp
-  | apply iff
-  | intro;
-    split;
-    intro;
-    assumption
-  | intros 3;
-    cases H; clear H;
-    split;
-    assumption
-  | intros 5;
-    cases H; cases H1; clear H H1;
-    split;
-    intros;
-    [ apply (H4 (H2 H))
-    | apply (H3 (H5 H))]]
-qed.
-
-record function_space1 (A: setoid1) (B: setoid1): Type ≝
- { f1:1> A → B;
-   f1_ok: ∀a,a':A. proofs1 (eq1 ? a a') → proofs1 (eq1 ? (f1 a) (f1 a'))
- }.
-
-definition function_space_setoid1: setoid1 → setoid1 → setoid1.
- intros (A B);
- constructor 1;
-  [ apply (function_space1 A B);
-  | intros;
-    apply (∀a:A. proofs1 (eq1 ? (f a) (f1 a)));
-  |*: cases daemon] (* simplify;
-    intros;
-    apply (f1_ok ? ? x);
-    unfold proofs; simplify;
-    apply (refl1 A)
-  | simplify;
-    intros;
-    unfold proofs; simplify;
-    apply (sym1 B);
-    apply (f a)
-  | simplify;
-    intros;
-    unfold carr; unfold proofs; simplify;
-    apply (trans1 B ? (y a));
-    [ apply (f a)
-    | apply (f1 a)]] *)
-qed.
-
-interpretation "function_space_setoid1" 'Imply a b = (function_space_setoid1 a b).
-
-definition setoids: setoid1.
- constructor 1;
-  [ apply setoid;
-  | apply isomorphism;
-  | intro;
-    split;
-     [1,2: constructor 1;
-        [1,3: intro; assumption;
-        |*: intros; assumption]
-     |3,4:
-       intros;
-       simplify;
-       unfold proofs; simplify;
-       apply refl;]
-  |*: cases daemon]
-qed.
-
-definition setoid1_of_setoid: setoid → setoid1.
- intro;
- constructor 1;
-  [ apply (carr s)
-  | apply (eq s)
-  | apply (refl s)
-  | apply (sym s)
-  | apply (trans s)]
-qed.
-
-coercion setoid1_of_setoid.
-
-record forall (A:setoid)  (B: A ⇒ CCProp): CProp ≝
- { fo:> ∀a:A.proofs (B a) }.
-
-record subset (A: setoid) : CProp ≝
- { mem: A ⇒ CCProp }.
-
-definition ssubset: setoid → setoid1.
- intro;
- constructor 1;
-  [ apply (subset s);
-  | apply (λU,V:subset s. ∀a. mem ? U a \liff mem ? V a)
-  | simplify;
-    intros;
-    split;
-    intro;
-    assumption
-  | simplify;
-    cases daemon
-  | cases daemon]
-qed.
-
-definition mmem: ∀A:setoid. (ssubset A) ⇒ A ⇒ CCProp.
- intros;
- constructor 1;
-  [ apply mem; 
-  | unfold function_space_setoid1; simplify;
-    intros (b b');
-    change in ⊢ (? (? (?→? (? %)))) with (mem ? b a \liff mem ? b' a);
-    unfold proofs1; simplify; intros;
-    unfold proofs1 in c; simplify in c;
-    unfold ssubset in c; simplify in c;
-    cases (c a); clear c;
-    split;
-    assumption]
-qed.
-
-definition sand: CCProp ⇒ CCProp.
-
-definition intersection: ∀A. ssubset A ⇒ ssubset A ⇒ ssubset A.
- intro;
- constructor 1;
-  [ intro;
-    constructor 1;
-     [ intro;
-       constructor 1;
-       constructor 1;
-       intro;
-       apply (mem ? c c2 ∧ mem ? c1 c2);
-     |
-  |
-  |
-
-*)
+include "hints_declaration.ma".
+
+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; @ (carr s); @ (eq0…) (refl…) (sym…) (trans…);
+nqed.
+
+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 ? (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 ? (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 ".=_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') 
+}.
+ 
+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 "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 ≡ unary_morphism1 S 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 ??? (fun11 ?? f) (fun11 ?? g))
+        (prop11 ?? (comp1_unary_morphisms o1 o2 o3 f g)))
+ (* -------------------------------------------------------------------- *) ⊢
+       fun11 o1 o3 R ≡ composition1 ??? (fun11 ?? f) (fun11 ?? 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.