X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fsets.ma;h=c32d194363d282e57148a028257b1a215f230ba6;hb=1144eafda5f046c21ceda807f4b5659b3e74147d;hp=f140ef7700b7ca3b426debca3612628d8ac7f876;hpb=b266ed97b63400d62ab4ba6a4ebdfbc1d5b0c2bb;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/sets.ma b/helm/software/matita/nlibrary/sets/sets.ma
index f140ef770..c32d19436 100644
--- a/helm/software/matita/nlibrary/sets/sets.ma
+++ b/helm/software/matita/nlibrary/sets/sets.ma
@@ -71,7 +71,7 @@ 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 ********************)
 
@@ -83,13 +83,21 @@ nrecord qpowerclass (A: setoid) : Type[1] ≝
                 ma la sintassi :> non lo supporta *)
    mem_ok': ∀x,x':A. x=x' → (x ∈ pc) = (x' ∈ pc) 
  }.
+ 
+notation > "𝛀 ^ term 90 A" non associative with precedence 70 
+for @{ 'qpowerclass $A }.
 
-ndefinition Full_set: ∀A. qpowerclass A.
+notation "Ω term 90 A \atop ≈" non associative with precedence 70 
+for @{ 'qpowerclass $A }.
+
+interpretation "qpowerclass" 'qpowerclass a = (qpowerclass a).
+
+ndefinition Full_set: ∀A. 𝛀^A.
  #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).
+ndefinition qseteq: ∀A. equivalence_relation1 (𝛀^A).
  #A; @
   [ napply (λS,S'. S = S')
   | #S; napply (refl1 ? (seteq A))
@@ -102,9 +110,13 @@ ndefinition qpowerclass_setoid: setoid → setoid1.
   [ napply (qpowerclass A)
   | napply (qseteq A) ]
 nqed.
-
+              
 unification hint 0 ≔ A ⊢  
-  carr1 (qpowerclass_setoid A) ≡ qpowerclass A.
+  carr1 (mk_setoid1 (𝛀^A) (eq1 (qpowerclass_setoid A))) 
+≡ qpowerclass A.
+
+ncoercion pc' : ∀A.∀x:qpowerclass_setoid A. Ω^A ≝ pc 
+on _x : (carr1 (qpowerclass_setoid ?)) to (Ω^?). 
 
 nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) CPROP.
  #A; @
@@ -116,9 +128,16 @@ nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid
   ##]
 nqed.
 
-unification hint 0 ≔ 
-  A : setoid, x, S ⊢ (mem_ok A) x S ≡ mem A S x.
-  
+unification hint 0 ≔  A:setoid, x, S;  
+         SS ≟ (pc ? S),
+         TT ≟ (mk_binary_morphism1 ??? 
+                 (λx:setoid1_of_setoid ?.λS:qpowerclass_setoid ?. x ∈ S) 
+                 (prop21 ??? (mem_ok A)))
+           
+  (*-------------------------------------*) ⊢ 
+    fun21 ? ? ? TT x S 
+  ≡ mem A SS x.
+
 nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) CPROP.
  #A; @
   [ napply (λS,S'. S ⊆ S')
@@ -129,15 +148,88 @@ nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_s
         [ 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. 𝛀^A → 𝛀^A → 𝛀^A.
+ #A; #S; #S'; @ (S ∩ S');
+ #a; #a'; #Ha; @; *; #H1; #H2; @
+  [##1,2: napply (. Ha^-1‡#); nassumption;
+##|##3,4: napply (. Ha‡#); nassumption]
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 1 ≔ 
+  A : setoid, B,C : qpowerclass A ⊢ 
+    pc A (mk_qpowerclass ? (B ∩ C) (mem_ok' ? (intersect_ok ? B C))) 
+    ≡ intersect ? (pc ? B) (pc ? C).
+
+nlemma intersect_ok': ∀A. binary_morphism1 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A).
+ #A; @ (λS,S'. S ∩ S');
+ #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *; #Ka; #Kb; @
+  [ napply Ha1; nassumption
+  | napply Hb1; nassumption
+  | napply Ha2; nassumption
+  | napply Hb2; nassumption]
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔ 
+  A : Type[0], B,C : powerclass A ⊢ 
+    fun21 …
+     (mk_binary_morphism1 …
+       (λS,S'.S ∩ S') 
+       (prop21 … (intersect_ok' A))) B C
+    ≡ intersect ? B C.
+
+ndefinition prop21_mem : 
+  ∀A,C.∀f:binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) C.
+   ∀a,a':setoid1_of_setoid A.
+    ∀b,b':qpowerclass_setoid A.a = a' → b = b' → f a b = f a' b'.
+#A; #C; #f; #a; #a'; #b; #b'; #H1; #H2; napply prop21; nassumption;
+nqed.
+    
+interpretation "prop21 mem" 'prop2 l r = (prop21_mem ??????? l r).
+
+nlemma intersect_ok'': 
+  ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
+ #A; @ (intersect_ok A); nlapply (prop21 … (intersect_ok' A)); #H;
+ #a; #a'; #b; #b'; #H1; #H2; napply H; nassumption; 
+nqed.
+
+unification hint 1 ≔ 
+  A:?, B,C : 𝛀^A ⊢ 
+    fun21 …
+     (mk_binary_morphism1 …
+       (λS,S':qpowerclass_setoid A.S ∩ S') 
+       (prop21 … (intersect_ok'' A))) B C
+    ≡ intersect ? B C.
+
+
+    
+    
+nlemma test: ∀U.∀A,B:qpowerclass U. A ∩ B = A →
+ ∀x,y. x=y → x ∈ A → y ∈ A ∩ B.
+ #U; #A; #B; #H; #x; #y; #K; #K2; napply (. #‡(?));
+##[ nchange with (A ∩ B = ?);
+    napply (prop21 ??? (mk_binary_morphism1 … (λS,S'.S ∩ S') (prop21 … (intersect_ok' U))) A A B B ##);
+    #H; napply H;
+  nassumption;
+nqed. 
+
+(*
 nlemma intersect_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
  #A; @
   [ #S; #S'; @
      [ napply (S ∩ S')
-     | #a; #a'; #Ha; nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
-        [##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'; #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
-      [ napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
+      [ alias symbol "invert" = "setoid1 symmetry".
+        napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
       | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
 nqed.
 
@@ -145,17 +237,18 @@ nqed.
 alias symbol "hint_decl" = "hint_decl_Type1".
 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.
+    pc A (fun21 …
+            (mk_binary_morphism1 …
+              (λS,S':qpowerclass_setoid A.mk_qpowerclass ? (S ∩ S') (mem_ok' ? (intersect_ok ? S S'))) 
+              (prop21 … (intersect_ok A))) 
+            B
+            C) 
+    ≡ intersect ? (pc ? B) (pc ? C).
 
 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.
+*)
 
 ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
  λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
@@ -169,10 +262,6 @@ 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.
@@ -187,7 +276,8 @@ ndefinition eqrel_of_morphism:
   [ @
      [ 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).
@@ -219,7 +309,7 @@ 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; @ y; @ [ napply I | napply refl] 
+ #A; #B; #f; nwhd; #y; #Hy; @ y; @ I ; napply refl; 
  (* bug, prova @ I refl *)
 nqed.
 
@@ -229,9 +319,31 @@ 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 ] ).   
+   
+   ;
+ }.
+*)