X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fsets.ma;h=c4fc89f210b0c90f5a049d80442f0cf6b7222e11;hb=8cb2490b5b202549a596cfd1d0f166a5ee43fc4e;hp=68a281500480d917106be8d0495182123dcfff29;hpb=bc4cc41e813fc850a7f981cd60ba22e14485b7d1;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/sets.ma b/helm/software/matita/nlibrary/sets/sets.ma
index 68a281500..c4fc89f21 100644
--- a/helm/software/matita/nlibrary/sets/sets.ma
+++ b/helm/software/matita/nlibrary/sets/sets.ma
@@ -39,7 +39,6 @@ ndefinition big_union ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∃i. i ∈ T ∧
 ndefinition big_intersection ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∀i. i ∈ T → x ∈ f i }.
 
 ndefinition full_set: ∀A. Ω^A ≝ λA.{ x | True }.
-ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
 
 nlemma subseteq_refl: ∀A.∀S: Ω^A. S ⊆ S.
  #A; #S; #x; #H; nassumption.
@@ -52,26 +51,27 @@ nqed.
 include "properties/relations1.ma".
 
 ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
- #A; napply mk_equivalence_relation1
+ #A; @
   [ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
-  | #S; napply conj; napply subseteq_refl
-  | #S; #S'; *; #H1; #H2; napply conj; nassumption
-  | #S; #T; #U; *; #H1; #H2; *; #H3; #H4; napply conj; napply subseteq_trans;
+  | #S; @; napply subseteq_refl
+  | #S; #S'; *; #H1; #H2; @; nassumption
+  | #S; #T; #U; *; #H1; #H2; *; #H3; #H4; @; napply subseteq_trans;
      ##[##2,5: nassumption |##1,4: ##skip |##*: nassumption]##]
-nqed. 
+nqed.
 
 include "sets/setoids1.ma".
 
+(* this has to be declared here, so that it is combined with carr *)
+ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
+
 ndefinition powerclass_setoid: Type[0] → setoid1.
- #A; napply mk_setoid1
-  [ napply (Ω^A)
-  | napply seteq ]
+ #A; @[ napply (Ω^A)| napply seteq ]
 nqed.
 
 include "hints_declaration.ma". 
 
 alias symbol "hint_decl" = "hint_decl_Type2".
-unification hint 0 ≔ A ⊢ carr1 (powerclass_setoid A) ≡ Ω^A.
+unification hint 0 ≔ A ⊢ carr1 (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A))) ≡ Ω^A.
 
 (************ SETS OVER SETOIDS ********************)
 
@@ -85,13 +85,12 @@ nrecord qpowerclass (A: setoid) : Type[1] ≝
  }.
 
 ndefinition Full_set: ∀A. qpowerclass A.
- #A; napply mk_qpowerclass
-  [ napply (full_set A)
-  | #x; #x'; #H; napply refl1; ##]
+ #A; @[ napply A | #x; #x'; #H; napply refl1]
 nqed.
+ncoercion Full_set: ∀A. qpowerclass A ≝ Full_set on A: setoid to qpowerclass ?.
 
 ndefinition qseteq: ∀A. equivalence_relation1 (qpowerclass A).
- #A; napply mk_equivalence_relation1
+ #A; @
   [ napply (λS,S'. S = S')
   | #S; napply (refl1 ? (seteq A))
   | #S; #S'; napply (sym1 ? (seteq A))
@@ -99,7 +98,7 @@ ndefinition qseteq: ∀A. equivalence_relation1 (qpowerclass A).
 nqed.
 
 ndefinition qpowerclass_setoid: setoid → setoid1.
- #A; napply mk_setoid1
+ #A; @
   [ napply (qpowerclass A)
   | napply (qseteq A) ]
 nqed.
@@ -107,38 +106,55 @@ nqed.
 unification hint 0 ≔ A ⊢  
   carr1 (qpowerclass_setoid A) ≡ qpowerclass A.
 
+(*CSC: non va!  
+unification hint 0 ≔ A ⊢  
+  carr1 (mk_setoid1 (qpowerclass A) (eq1 (qpowerclass_setoid A))) ≡ qpowerclass A.*)
+
 nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) CPROP.
- #A; napply mk_binary_morphism1
+ #A; @
   [ napply (λx,S. x ∈ S) 
-  | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; napply mk_iff; #H;
+  | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H;
      ##[ napply Hb1; napply (. (mem_ok' …)); ##[##3: napply H| napply Ha^-1;##]
      ##| napply Hb2; napply (. (mem_ok' …)); ##[##3: napply H| napply Ha;##]
      ##]
   ##]
 nqed.
 
-unification hint 0 ≔ 
-  A : setoid, x, S ⊢ (mem_ok A) x S ≡ mem A S x.
-  
+(*CSC: bug qui se metto x o S al posto di ?
+nlemma foo: True.
+nletin xxx ≝ (λA:setoid.λx,S. let SS ≝ pc ? S in
+    fun21 ??? (mk_binary_morphism1 ??? (λx.λS. ? ∈ ?) (prop21 ??? (mem_ok A))) x S);
+*)
+unification hint 0 ≔  A:setoid, x, S;  
+         SS ≟ (pc ? S)
+  (*-------------------------------------*) ⊢ 
+    fun21 ??? (mk_binary_morphism1 ??? (λx,S. x ∈ S) (prop21 ??? (mem_ok A))) x S ≡ mem A SS x.
+
 nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) CPROP.
- #A; napply mk_binary_morphism1
+ #A; @
   [ napply (λS,S'. S ⊆ S')
-  | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; napply mk_iff; #H
+  | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #H
      [ napply (subseteq_trans … a)
         [ nassumption | napply (subseteq_trans … b); nassumption ]
    ##| napply (subseteq_trans … a')
         [ nassumption | napply (subseteq_trans … b'); nassumption ] ##]
 nqed.
 
+unification hint 0 ≔ A,a,a'
+ (*-----------------------------------------------------------------*) ⊢
+  eq_rel ? (eq A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
+
 nlemma intersect_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
- #A; napply mk_binary_morphism1
-  [ #S; #S'; napply mk_qpowerclass
+ #A; @
+  [ #S; #S'; @
      [ napply (S ∩ S')
-     | #a; #a'; #Ha; nwhd in ⊢ (? ? ? % %); napply mk_iff; *; #H1; #H2; napply conj
-        [##1,2: napply (. (mem_ok' …)^-1) [##3,6: nassumption |##2,5: nassumption |##*: ##skip]
-      ##|##3,4: napply (. (mem_ok' …)) [##1,3,4,6: nassumption |##*: ##skip]##]##]
- ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; napply conj; #x; nwhd in ⊢ (% → %); #H
-      [ napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
+     | #a; #a'; #Ha;
+        nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
+        [##1,2: napply (. Ha^-1‡#); nassumption;
+      ##|##3,4: napply (. Ha‡#); nassumption]##]
+ ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; @; #x; nwhd in ⊢ (% → %); #H
+      [ alias symbol "invert" = "setoid1 symmetry".
+        napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
       | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
 nqed.
 
@@ -148,12 +164,6 @@ unification hint 0 ≔
   A : setoid, B,C : qpowerclass A ⊢ 
     pc A (intersect_ok A B C) ≡ intersect ? (pc ? B) (pc ? C).
 
-(* hints can pass under mem *) (* ??? XXX why is it needed? *)
-unification hint 0 ≔ A,B,x ;
-           C ≟ B
- (*---------------------*) ⊢ 
-   mem A B x ≡ mem A C x.
-
 nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
  #A; #U; #V; #x; #x'; #H; #p; napply (. (H^-1‡#)); nassumption.
 nqed.
@@ -170,38 +180,33 @@ ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
 nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
  { rel:> equivalence_relation A;
    compatibility: ∀x,x':A. x=x' → rel x x'
-    (* coercion qui non andava per via di un Failure invece di Uncertain
-       ritornato dall'unificazione per il problema: 
-         ?[] A =?= ?[Γ]->?[Γ+1] 
-    *)
  }.
 
 ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
- #A; #R; napply mk_setoid
-  [ napply A
-  | napply R]
+ #A; #R; @ A R; 
 nqed.
 
 (******************* first omomorphism theorem for sets **********************)
 
 ndefinition eqrel_of_morphism:
  ∀A,B. unary_morphism A B → compatible_equivalence_relation A.
- #A; #B; #f; napply mk_compatible_equivalence_relation
-  [ napply mk_equivalence_relation
+ #A; #B; #f; @
+  [ @
      [ napply (λx,y. f x = f y)
      | #x; napply refl | #x; #y; napply sym | #x; #y; #z; napply trans]
-##| #x; #x'; #H; nwhd; napply (.= (†H)); napply refl ]
+##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
+napply (.= (†H)); napply refl ]
 nqed.
 
 ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
- #A; #R; napply mk_unary_morphism
+ #A; #R; @
   [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
 nqed.
 
 ndefinition quotiented_mor:
  ∀A,B.∀f:unary_morphism A B.
   unary_morphism (quotient … (eqrel_of_morphism … f)) B.
- #A; #B; #f; napply mk_unary_morphism
+ #A; #B; #f; @
   [ napply f | #a; #a'; #H; nassumption]
 nqed.
 
@@ -222,8 +227,8 @@ ndefinition injective ≝
 nlemma first_omomorphism_theorem_functions2:
  ∀A,B.∀f: unary_morphism A B. 
    surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
- #A; #B; #f; nwhd; #y; #Hy; napply (ex_intro … y); napply conj
-  [ napply I | napply refl]
+ #A; #B; #f; nwhd; #y; #Hy; @ y; @ I ; napply refl; 
+ (* bug, prova @ I refl *)
 nqed.
 
 nlemma first_omomorphism_theorem_functions3:
@@ -232,9 +237,31 @@ nlemma first_omomorphism_theorem_functions3:
  #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
 nqed.
 
-nrecord isomorphism (A) (B) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
+nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : Type[0] ≝
  { iso_f:> unary_morphism A B;
    f_closed: ∀x. x ∈ S → iso_f x ∈ T;
    f_sur: surjective … S T iso_f;
    f_inj: injective … S iso_f
  }.
+
+(*
+nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
+ { iso_f:> unary_morphism A B;
+   f_closed: ∀x. x ∈ pc A S → fun1 ?? iso_f x ∈ pc B T}.
+   
+   
+ncheck (λA:?.
+   λB:?.
+    λS:?.
+     λT:?.
+      λxxx:isomorphism A B S T.
+       match xxx
+       return λxxx:isomorphism A B S T.
+               ∀x: carr A.
+                ∀x_72: mem (carr A) (pc A S) x.
+                 mem (carr B) (pc B T) (fun1 A B ((λ_.?) A B S T xxx) x)
+        with [ mk_isomorphism _ yyy ⇒ yyy ] ).   
+   
+   ;
+ }.
+*)