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[helm.git] / helm / software / matita / nlibrary / sets / sets.ma

diff --git a/helm/software/matita/nlibrary/sets/sets.ma b/helm/software/matita/nlibrary/sets/sets.ma
index c4fc89f210b0c90f5a049d80442f0cf6b7222e11..96521773738786bc15cd6eb0d2bbc4c35ce8be50 100644 (file)
--- a/helm/software/matita/nlibrary/sets/sets.ma
+++ b/helm/software/matita/nlibrary/sets/sets.ma
@@ -102,13 +102,13 @@ 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 (qpowerclass A) (eq1 (qpowerclass_setoid A))) 
+≡ qpowerclass A.
 
-(*CSC: non va!  
-unification hint 0 ≔ A ⊢  
-  carr1 (mk_setoid1 (qpowerclass 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; @
@@ -120,15 +120,15 @@ nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid
   ##]
 nqed.
 
-(*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)
+         SS ≟ (pc ? S),
+         TT ≟ (mk_binary_morphism1 ??? 
+                 (λx:setoid1_of_setoid ?.λS:qpowerclass_setoid ?. x ∈ S) 
+                 (prop21 ??? (mem_ok A)))
+           
   (*-------------------------------------*) ⊢ 
-    fun21 ??? (mk_binary_morphism1 ??? (λx,S. x ∈ S) (prop21 ??? (mem_ok A))) x S ≡ mem A SS x.
+    fun21 ? ? ? TT x S 
+  ≡ mem A SS x.
 
 nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) CPROP.
  #A; @
@@ -144,6 +144,53 @@ 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. qpowerclass A → qpowerclass A → qpowerclass 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 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 (. K^-1‡H); nassumption;
+nqed. 
+
+(*
 nlemma intersect_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
  #A; @
   [ #S; #S'; @
@@ -162,11 +209,18 @@ 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).
+    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.