X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fsets.ma;h=4e7418fd93bcc17f4777d3bb30a293d674bc43c3;hb=6fe06927f3293bfce4a01a587abd9913e711da88;hp=f4b924dccaa26dc047f8ebc557d909b41f4e01fe;hpb=3807c5d0fa5abceccbc67f40edb9939b353ead0e;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/sets.ma b/helm/software/matita/nlibrary/sets/sets.ma
index f4b924dcc..4e7418fd9 100644
--- a/helm/software/matita/nlibrary/sets/sets.ma
+++ b/helm/software/matita/nlibrary/sets/sets.ma
@@ -12,105 +12,247 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "logic/cprop.ma".
+(******************* SETS OVER TYPES *****************)
+
+include "logic/connectives.ma".
 
-nrecord powerset (A: setoid) : Type[1] ≝ { mem_op:> unary_morphism1 A CPROP }.
+nrecord powerclass (A: Type[0]) : Type[1] ≝ { mem: A → CProp[0] }.
 
-interpretation "powerset" 'powerset A = (powerset A).
+interpretation "mem" 'mem a S = (mem ? S a).
+interpretation "powerclass" 'powerset A = (powerclass A).
+interpretation "subset construction" 'subset \eta.x = (mk_powerclass ? x).
 
-interpretation "subset construction" 'subset \eta.x =
- (mk_powerset ? (mk_unary_morphism1 ? CPROP x ?)).
+ndefinition subseteq ≝ λA.λU,V.∀a:A. a ∈ U → a ∈ V.
+interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
 
-interpretation "mem" 'mem a S = (mem_op ? S a).
+ndefinition overlaps ≝ λA.λU,V.∃x:A.x ∈ U ∧ x ∈ V.
+interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
 
-ndefinition subseteq ≝ λA:setoid.λU,V.∀a:A. a ∈ U → a ∈ V.
+ndefinition intersect ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∧ x ∈ V }.
+interpretation "intersect" 'intersects U V = (intersect ? U V).
 
-interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
+ndefinition union ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∨ x ∈ V }.
+interpretation "union" 'union U V = (union ? U V).
 
-ntheorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
- #A; #S; #x; #H; nassumption;
+ndefinition big_union ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∃i. i ∈ T ∧ x ∈ f i }.
+
+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 }.
+
+nlemma subseteq_refl: ∀A.∀S: Ω^A. S ⊆ S.
+ #A; #S; #x; #H; nassumption.
 nqed.
 
-ntheorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
- #A; #S1; #S2; #S3; #H12; #H23; #x; #H;
- napply H23; napply H12; nassumption;
+nlemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
+ #A; #S; #T; #U; #H1; #H2; #x; #P; napply H2; napply H1; nassumption.
 nqed.
 
-ndefinition powerset_setoid1: setoid → setoid1.
- #S; napply mk_setoid1
-  [ napply (Ω \sup S)
-  | napply mk_equivalence_relation1
-     [ #A; #B; napply (∀x. iff (x ∈ A) (x ∈ B))
-     | nwhd; #x; #x0; napply mk_iff; #H; nassumption
-     | nwhd; #x; #y; #H; #A; napply mk_iff; #K
-        [ napply (fi ?? (H ?)) | napply (if ?? (H ?)) ]
-        nassumption
-     | nwhd; #A; #B; #C; #H1; #H2; #H3; napply mk_iff; #H4
-        [ napply (if ?? (H2 ?)); napply (if ?? (H1 ?)); nassumption
-        | napply (fi ?? (H1 ?)); napply (fi ?? (H2 ?)); nassumption]##]
+include "properties/relations1.ma".
+
+ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
+ #A; @
+  [ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
+  | #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.
 
-unification hint 0 (∀A.(λx,y.True) (Ω \sup A) (carr1 (powerset_setoid1 A))).
+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 mem: ∀A:setoid. binary_morphism1 A (powerset_setoid1 A) CPROP.
- #A; napply mk_binary_morphism1
-  [ napply (λa.λA.a ∈ A)
-  | #a; #a'; #B; #B'; #Ha; #HB; napply mk_iff; #H
-     [ napply (. (†Ha^-1)); (* CSC: notation for ∈ not working *)
-       napply (if ?? (HB ?)); nassumption
-     | napply (. (†Ha)); napply (fi ?? (HB ?)); nassumption]##]
+ndefinition powerclass_setoid: Type[0] → setoid1.
+ #A; @[ napply (Ω^A)| napply seteq ]
 nqed.
 
-unification hint 0 (∀A,x,S. (λx,y.True) (mem_op A x S) (fun21 ??? (mem A) S x)).
+include "hints_declaration.ma". 
 
-ndefinition overlaps ≝ λA.λU,V:Ω \sup A.∃x:A.x ∈ U ∧ x ∈ V.
+alias symbol "hint_decl" = "hint_decl_Type2".
+unification hint 0 ≔ A ⊢ carr1 (powerclass_setoid A) ≡ Ω^A.
 
-interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
+(************ SETS OVER SETOIDS ********************)
 
-ndefinition intersects ≝ λA:Type[0].λU,V:A → CProp[0]. λx. U x ∧ V x.
+include "logic/cprop.ma".
 
-interpretation "intersects" 'intersects U V = (intersects ? U V).
+nrecord qpowerclass (A: setoid) : Type[1] ≝
+ { pc:> Ω^A; (* qui pc viene dichiarato con un target preciso... 
+                forse lo si vorrebbe dichiarato con un target più lasco 
+                ma la sintassi :> non lo supporta *)
+   mem_ok': ∀x,x':A. x=x' → (x ∈ pc) = (x' ∈ pc) 
+ }.
 
-(* dovrebbe essere un binario? *)
-ndefinition intersects_ok: ∀A. Ω \sup A → Ω \sup A → Ω \sup A.
- #A; #U; #V; napply mk_powerset; napply mk_unary_morphism1
-  [ napply (intersects ? (mem_op ? U) (mem_op ? V))
-  | #a; #a'; #H; napply mk_iff; *; #H1; #H2
-    [ nwhd; napply (. ((H^-1‡#)‡(H^-1‡#))); nwhd; napply conj; nassumption
-    | nwhd; napply (. ((H‡#)‡(H‡#))); nwhd; napply conj; nassumption]
+ndefinition Full_set: ∀A. qpowerclass A.
+ #A; @[ napply A | #x; #x'; #H; napply refl1]
 nqed.
+ncoercion Full_set: ∀A. qpowerclass A ≝ Full_set on A: setoid to qpowerclass ?.
 
-unification hint 0 (∀A.∀U,V: Ω \sup A.∀w.(λx,y.True)
- (intersects A U V w) (fun11 ?? (mem_op ? (intersects_ok A U V)) w)).
+ndefinition qseteq: ∀A. equivalence_relation1 (qpowerclass A).
+ #A; @
+  [ napply (λS,S'. S = S')
+  | #S; napply (refl1 ? (seteq A))
+  | #S; #S'; napply (sym1 ? (seteq A))
+  | #S; #T; #U; napply (trans1 ? (seteq A))]
+nqed.
 
-nlemma test: ∀A. ∀U,V: Ω \sup A. ∀x,x':A. x=x' → (U ∩ V) x → (U ∩ V) x'.
- #A; #U; #V; #x; #x'; #H; #p;
- nwhd in ⊢ (? ? % % ?);
- (* l'unification hint non funziona *)
-  nchange with (? ∈ (intersects_ok ? ? ?));
-  napply (. (†H^-1));
-  nassumption.
+ndefinition qpowerclass_setoid: setoid → setoid1.
+ #A; @
+  [ napply (qpowerclass A)
+  | napply (qseteq A) ]
 nqed.
 
-(*
-ndefinition union ≝ λA.λU,V:Ω \sup A. {x | x ∈ U ∨ x ∈ V }.
+unification hint 0 ≔ A ⊢  
+  carr1 (qpowerclass_setoid A) ≡ qpowerclass A.
 
-interpretation "union" 'union U V = (union ? U V).
+nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) CPROP.
+ #A; @
+  [ napply (λx,S. x ∈ S) 
+  | #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;  
+         SS ≟ (pc ? S)
+  (*-------------------------------------*) ⊢ 
+    fun21 ??? (mem_ok A) x S ≡ mem A SS x.
 
-ndefinition singleton ≝ λA:setoid.λa:A.{b | a=b}.
+nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) CPROP.
+ #A; @
+  [ napply (λS,S'. S ⊆ S')
+  | #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.
 
-interpretation "singleton" 'singl a = (singleton ? a).*)
+unification hint 0 ≔ A,a,a'
+ (*-----------------------------------------------------------------*) ⊢
+  eq_rel ? (eq A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
 
-(*
-(* qui non funziona una cippa *)
-ndefinition image: ∀A,B. (carr A → carr B) → Ω \sup A → Ω \sup B ≝
- λA,B:setoid.λf:carr A → carr B.λSa:Ω \sup A.
-  {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) ? ?(*(f x) y*)}.
-  ##[##2: napply (f x); ##|##3: napply y]
- #a; #a'; #H; nwhd; nnormalize; (* per togliere setoid1_of_setoid *) napply (mk_iff ????);
- *; #x; #Hx; napply (ex_intro … x)
-  [ napply (. (#‡(#‡#))); 
-
-ndefinition counter_image: ∀A,B. (A → B) → Ω \sup B → Ω \sup A ≝
+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 (. 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.
+
+(* unfold if intersect, exposing fun21 *)
+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).
+
+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.
+  {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) (f x) y}.
+
+ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
  λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
+
+(******************* compatible equivalence relations **********************)
+
+nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
+ { rel:> equivalence_relation A;
+   compatibility: ∀x,x':A. x=x' → rel x x'
+ }.
+
+ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
+ #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 (λx,y. f x = f y)
+     | #x; napply refl | #x; #y; napply sym | #x; #y; #z; napply trans]
+##| #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 (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 f | #a; #a'; #H; nassumption]
+nqed.
+
+nlemma first_omomorphism_theorem_functions1:
+ ∀A,B.∀f: unary_morphism A B.
+  ∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
+ #A; #B; #f; #x; napply refl;
+nqed.
+
+ndefinition surjective ≝
+ λA,B.λS: qpowerclass A.λT: qpowerclass B.λf:unary_morphism A B.
+  ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
+
+ndefinition injective ≝
+ λA,B.λS: qpowerclass A.λf:unary_morphism A B.
+  ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
+
+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; @ I ; napply refl; 
+ (* bug, prova @ I refl *)
+nqed.
+
+nlemma first_omomorphism_theorem_functions3:
+ ∀A,B.∀f: unary_morphism A B. 
+   injective … (Full_set ?) (quotiented_mor … f).
+ #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
+nqed.
+
+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 ] ).   
+   
+   ;
+ }.
 *)