X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Ftopology%2Figft.ma;h=7f4f5e3591724cfef8502d4659fbc1e3574a89c9;hb=e956eab1116ae48a298e3c6701f93178e53ab24f;hp=29e1f1af31995ddc7f1a17158b3ee6c07fcd8d8d;hpb=5bfea0766a39d05605b6e88508f5376a88351331;p=helm.git

diff --git a/helm/software/matita/nlibrary/topology/igft.ma b/helm/software/matita/nlibrary/topology/igft.ma
index 29e1f1af3..7f4f5e359 100644
--- a/helm/software/matita/nlibrary/topology/igft.ma
+++ b/helm/software/matita/nlibrary/topology/igft.ma
@@ -1,174 +1,301 @@
-include "logic/connectives.ma".
+include "sets/sets.ma".
 
-nrecord powerset (X : Type[0]) : Type[1] ≝ { char : X → CProp[0] }.
-
-interpretation "char" 'subset p = (mk_powerset ? p).  
-
-interpretation "pwset" 'powerset a = (powerset a). 
-
-interpretation "in" 'mem a X = (char ? X a). 
-
-ndefinition subseteq ≝ λA.λU,V : Ω^A. 
-  ∀x.x ∈ U → x ∈ V.
-
-interpretation "subseteq" 'subseteq u v = (subseteq ? u v).
-
-ndefinition overlaps ≝ λA.λU,V : Ω^A. 
-  ∃x.x ∈ U ∧ x ∈ V.
-
-interpretation "overlaps" 'overlaps u v = (overlaps ? u v).
-(*
-ndefinition intersect ≝ λA.λu,v:Ω\sup A.{ y | y ∈ u ∧ y ∈ v }.
-
-interpretation "intersect" 'intersects u v = (intersect ? u v). 
-*)
-nrecord axiom_set : Type[1] ≝ { 
-  S:> Type[0];
+nrecord Ax : Type[1] ≝ { 
+  S:> setoid; (* Type[0]; *)
   I: S → Type[0];
   C: ∀a:S. I a → Ω ^ S
 }.
 
-ndefinition cover_set ≝ λc:∀A:axiom_set.Ω^A → A → CProp[0].λA,C,U.
+notation "𝐈  \sub( ❨a❩ )" non associative with precedence 70 for @{ 'I $a }.
+notation "𝐂 \sub ( ❨a,\emsp i❩ )" non associative with precedence 70 for @{ 'C $a $i }.
+
+notation > "𝐈 term 90 a" non associative with precedence 70 for @{ 'I $a }.
+notation > "𝐂 term 90 a term 90 i" non associative with precedence 70 for @{ 'C $a $i }.
+
+interpretation "I" 'I a = (I ? a).
+interpretation "C" 'C a i = (C ? a i).
+
+ndefinition cover_set ≝ λc:∀A:Ax.Ω^A → A → CProp[0].λA,C,U.
   ∀y.y ∈ C → c A U y.
 
+(* a \ltri b *)
 notation "hvbox(a break ◃ b)" non associative with precedence 45
-for @{ 'covers $a $b }. (* a \ltri b *)
+for @{ 'covers $a $b }.
 
 interpretation "covers set temp" 'covers C U = (cover_set ?? C U).
 
-ninductive cover (A : axiom_set) (U : Ω^A) : A → CProp[0] ≝ 
+ninductive cover (A : Ax) (U : Ω^A) : A → CProp[0] ≝ 
 | creflexivity : ∀a:A. a ∈ U → cover ? U a
-| cinfinity    : ∀a:A. ∀i:I ? a. (C ? a i ◃ U) → cover ? U a.
+| cinfinity    : ∀a:A.∀i:𝐈 a. 𝐂 a i ◃ U → cover ? U a.
 napply cover;
 nqed.
 
 interpretation "covers" 'covers a U = (cover ? U a).
 interpretation "covers set" 'covers a U = (cover_set cover ? a U).
 
-ndefinition fish_set ≝ λf:∀A:axiom_set.Ω^A → A → CProp[0].
+ndefinition fish_set ≝ λf:∀A:Ax.Ω^A → A → CProp[0].
  λA,U,V.
   ∃a.a ∈ V ∧ f A U a.
 
+(* a \ltimes b *)
 notation "hvbox(a break ⋉ b)" non associative with precedence 45
-for @{ 'fish $a $b }. (* a \ltimes b *)
+for @{ 'fish $a $b }. 
 
 interpretation "fish set temp" 'fish A U = (fish_set ?? U A).
 
-ncoinductive fish (A : axiom_set) (F : Ω^A) : A → CProp[0] ≝ 
-| cfish : ∀a. a ∈ F → (∀i:I ? a.C A a i ⋉ F) → fish A F a.
+ncoinductive fish (A : Ax) (F : Ω^A) : A → CProp[0] ≝ 
+| cfish : ∀a. a ∈ F → (∀i:𝐈 a .𝐂  a i ⋉ F) → fish A F a.
 napply fish;
 nqed.
 
 interpretation "fish set" 'fish A U = (fish_set fish ? U A).
 interpretation "fish" 'fish a U = (fish ? U a).
 
-nlet corec fish_rec (A:axiom_set) (U: Ω^A)
+nlet corec fish_rec (A:Ax) (U: Ω^A)
  (P: Ω^A) (H1: P ⊆ U)
-  (H2: ∀a:A. a ∈ P → ∀j: I ? a. C ? a j ≬ P):
+  (H2: ∀a:A. a ∈ P → ∀j: 𝐈 a. 𝐂 a j ≬ P):
    ∀a:A. ∀p: a ∈ P. a ⋉ U ≝ ?.
 #a; #p; napply cfish;
 ##[ napply H1; napply p;
-##| #i; ncases (H2 a p i); #x; *; #xC; #xP; napply ex_intro; ##[napply x]
-    napply conj; ##[ napply xC ] napply (fish_rec ? U P); nassumption;
+##| #i; ncases (H2 a p i); #x; *; #xC; #xP; @; ##[napply x]
+    @; ##[ napply xC ] napply (fish_rec ? U P); nassumption;
+##]
+nqed.
+
+notation "◃U" non associative with precedence 55
+for @{ 'coverage $U }.
+
+ndefinition coverage : ∀A:Ax.∀U:Ω^A.Ω^A ≝ λA,U.{ a | a ◃ U }.
+
+interpretation "coverage cover" 'coverage U = (coverage ? U).
+
+ndefinition cover_equation : ∀A:Ax.∀U,X:Ω^A.CProp[0] ≝  λA,U,X. 
+  ∀a.a ∈ X ↔ (a ∈ U ∨ ∃i:𝐈 a.∀y.y ∈ 𝐂 a i → y ∈ X).  
+
+ntheorem coverage_cover_equation : ∀A,U. cover_equation A U (◃U).
+#A; #U; #a; @; #H;
+##[ nelim H; #b; (* manca clear *)
+    ##[ #bU; @1; nassumption;
+    ##| #i; #CaiU; #IH; @2; @ i; #c; #cCbi; ncases (IH ? cCbi);
+        ##[ #E; @; napply E;
+        ##| #_; napply CaiU; nassumption; ##] ##]
+##| ncases H; ##[ #E; @; nassumption]
+    *; #j; #Hj; @2 j; #w; #wC; napply Hj; nassumption;
+##]
+nqed. 
+
+ntheorem coverage_min_cover_equation : 
+  ∀A,U,W. cover_equation A U W → ◃U ⊆ W.
+#A; #U; #W; #H; #a; #aU; nelim aU; #b;
+##[ #bU; ncases (H b); #_; #H1; napply H1; @1; nassumption;
+##| #i; #CbiU; #IH; ncases (H b); #_; #H1; napply H1; @2; @i; napply IH;
+##]
+nqed.
+
+notation "⋉F" non associative with precedence 55
+for @{ 'fished $F }.
+
+ndefinition fished : ∀A:Ax.∀F:Ω^A.Ω^A ≝ λA,F.{ a | a ⋉ F }.
+
+interpretation "fished fish" 'fished F = (fished ? F).
+
+ndefinition fish_equation : ∀A:Ax.∀F,X:Ω^A.CProp[0] ≝ λA,F,X.
+  ∀a. a ∈ X ↔ a ∈ F ∧ ∀i:𝐈 a.∃y.y ∈ 𝐂 a i ∧ y ∈ X. 
+  
+ntheorem fised_fish_equation : ∀A,F. fish_equation A F (⋉F).
+#A; #F; #a; @; (* bug, fare case sotto diverso da farlo sopra *) #H; ncases H;
+##[ #b; #bF; #H2; @ bF; #i; ncases (H2 i); #c; *; #cC; #cF; @c; @ cC;
+    napply cF;  
+##| #aF; #H1; @ aF; napply H1;
+##]
+nqed.
+
+ntheorem fised_max_fish_equation : ∀A,F,G. fish_equation A F G → G ⊆ ⋉F.
+#A; #F; #G; #H; #a; #aG; napply (fish_rec … aG);
+#b; ncases (H b); #H1; #_; #bG; ncases (H1 bG); #E1; #E2; nassumption; 
+nqed. 
+
+nrecord nAx : Type[2] ≝ { 
+  nS:> setoid; (*Type[0];*)
+  nI: nS → Type[0];
+  nD: ∀a:nS. nI a → Type[0];
+  nd: ∀a:nS. ∀i:nI a. nD a i → nS
+}.
+
+(*
+TYPE f A → B, g : B → A, f ∘ g = id, g ∘ g = id.
+
+a = b → I a = I b
+*)
+
+notation "𝐃 \sub ( ❨a,\emsp i❩ )" non associative with precedence 70 for @{ 'D $a $i }.
+notation "𝐝 \sub ( ❨a,\emsp i,\emsp j❩ )" non associative with precedence 70 for @{ 'd $a $i $j}.
+
+notation > "𝐃 term 90 a term 90 i" non associative with precedence 70 for @{ 'D $a $i }.
+notation > "𝐝 term 90 a term 90 i term 90 j" non associative with precedence 70 for @{ 'd $a $i $j}.
+
+interpretation "D" 'D a i = (nD ? a i).
+interpretation "d" 'd a i j = (nd ? a i j).
+interpretation "new I" 'I a = (nI ? a).
+
+ndefinition image ≝ λA:nAx.λa:A.λi. { x | ∃j:𝐃 a i. x = 𝐝 a i j }.
+
+notation > "𝐈𝐦  [𝐝 term 90 a term 90 i]" non associative with precedence 70 for @{ 'Im $a $i }.
+notation "𝐈𝐦  [𝐝 \sub ( ❨a,\emsp i❩ )]" non associative with precedence 70 for @{ 'Im $a $i }.
+
+interpretation "image" 'Im a i = (image ? a i).
+
+ndefinition Ax_of_nAx : nAx → Ax.
+#A; @ A (nI ?); #a; #i; napply (𝐈𝐦 [𝐝 a i]);
+nqed.
+
+ninductive sigma (A : Type[0]) (P : A → CProp[0]) : Type[0] ≝ 
+ sig_intro : ∀x:A.P x → sigma A P. 
+
+interpretation "sigma" 'sigma \eta.p = (sigma ? p). 
+
+ndefinition nAx_of_Ax : Ax → nAx.
+#A; @ A (I ?);
+##[ #a; #i; napply (Σx:A.x ∈ 𝐂 a i);
+##| #a; #i; *; #x; #_; napply x;
+##]
+nqed.
+
+ninductive Ord (A : nAx) : Type[0] ≝ 
+ | oO : Ord A
+ | oS : Ord A → Ord A
+ | oL : ∀a:A.∀i.∀f:𝐃 a i → Ord A. Ord A.
+
+notation "Λ term 90 f" non associative with precedence 50 for @{ 'oL $f }.
+notation "x+1" non associative with precedence 50 for @{'oS $x }.
+
+interpretation "ordinals Lambda" 'oL f = (oL ? ? ? f).
+interpretation "ordinals Succ" 'oS x = (oS ? x).
+
+nlet rec famU (A : nAx) (U : Ω^A) (x : Ord A) on x : Ω^A ≝ 
+  match x with
+  [ oO ⇒ U
+  | oS y ⇒ let Un ≝ famU A U y in Un ∪ { x | ∃i.𝐈𝐦[𝐝 x i] ⊆ Un} 
+  | oL a i f ⇒ { x | ∃j.x ∈ famU A U (f j) } ].
+  
+notation < "term 90 U \sub (term 90 x)" non associative with precedence 50 for @{ 'famU $U $x }.
+notation > "U ⎽ term 90 x" non associative with precedence 50 for @{ 'famU $U $x }.
+
+interpretation "famU" 'famU U x = (famU ? U x).
+  
+ndefinition ord_coverage : ∀A:nAx.∀U:Ω^A.Ω^A ≝ λA,U.{ y | ∃x:Ord A. y ∈ famU ? U x }.
+
+ndefinition ord_cover_set ≝ λc:∀A:nAx.Ω^A → Ω^A.λA,C,U.
+  ∀y.y ∈ C → y ∈ c A U.
+
+interpretation "coverage new cover" 'coverage U = (ord_coverage ? U).
+interpretation "new covers set" 'covers a U = (ord_cover_set ord_coverage ? a U).
+interpretation "new covers" 'covers a U = (mem ? (ord_coverage ? U) a).
+
+ntheorem new_coverage_reflexive:
+  ∀A:nAx.∀U:Ω^A.∀a. a ∈ U → a ◃ U.
+#A; #U; #a; #H; @ (oO A); napply H;
+nqed.
+
+nlemma ord_subset:
+  ∀A:nAx.∀a:A.∀i,f,U.∀j:𝐃 a i.U⎽(f j) ⊆ U⎽(Λ f).
+#A; #a; #i; #f; #U; #j; #b; #bUf; @ j; nassumption;
+nqed.
+
+naxiom AC : ∀A,a,i,U.(∀j:𝐃 a i.∃x:Ord A.𝐝 a i j ∈ U⎽x) → (Σf.∀j:𝐃 a i.𝐝 a i j ∈ U⎽(f j)).
+
+naxiom setoidification :
+  ∀A:nAx.∀a,b:A.∀U.a=b → b ∈ U → a ∈ U.
+
+alias symbol "covers" = "new covers".
+alias symbol "covers" = "new covers set".
+alias symbol "covers" = "new covers".
+ntheorem new_coverage_infinity:
+  ∀A:nAx.∀U:Ω^A.∀a:A. (∃i:𝐈 a. 𝐈𝐦[𝐝 a i] ◃ U) → a ◃ U.
+#A; #U; #a; *; #i; #H; nnormalize in H;
+ncut (∀y:𝐃 a i.∃x:Ord A.𝐝 a i y ∈ U⎽x); ##[
+  #y; napply H; @ y; napply #; ##] #H'; 
+ncases (AC … H'); #f; #Hf;
+ncut (∀j.𝐝 a i j ∈ U⎽(Λ f));
+  ##[ #j; napply (ord_subset … f … (Hf j));##] #Hf';
+@ ((Λ f)+1); @2; nwhd; @i; #x; *; #d; #Hd; 
+napply (setoidification … Hd); napply Hf';
+nqed.
+
+(* move away *)
+nlemma subseteq_union: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
+#A; #U; #V; #W; #H; #H1; #x; *; #Hx; ##[ napply H; ##| napply H1; ##] nassumption;
+nqed. 
+
+nlemma new_coverage_min :  
+  ∀A:nAx.∀U:qpowerclass A.∀V.U ⊆ V → (∀a:A.∀i.𝐈𝐦[𝐝 a i] ⊆ V → a ∈ V) → ◃(pc ? U) ⊆ V.
+#A; #U; #V; #HUV; #Im;  #b; *; #o; ngeneralize in match b; nchange with ((pc ? U)⎽o ⊆ V);
+nelim o;
+##[ #b; #bU0; napply HUV; napply bU0;
+##| #p; #IH; napply subseteq_union; ##[ nassumption; ##]
+    #x; *; #i; #H; napply (Im ? i); napply (subseteq_trans … IH); napply H;
+##| #a; #i; #f; #IH; #x; *; #d; napply IH; ##]
+nqed.
+
+nlet rec famF (A: nAx) (F : Ω^A) (x : Ord A) on x : Ω^A ≝ 
+  match x with
+  [ oO ⇒ F
+  | oS o ⇒ let Fo ≝ famF A F o in Fo ∩ { x | ∀i:𝐈 x.∃j:𝐃 x i.𝐝 x i j ∈ Fo } 
+  | oL a i f ⇒ { x | ∀j:𝐃 a i.x ∈ famF A F (f j) }
+  ].
+
+interpretation "famF" 'famU U x = (famF ? U x).
+
+ndefinition ord_fished : ∀A:nAx.∀F:Ω^A.Ω^A ≝ λA,F.{ y | ∀x:Ord A. y ∈ F⎽x }.
+
+interpretation "fished new fish" 'fished U = (ord_fished ? U).
+interpretation "new fish" 'fish a U = (mem ? (ord_fished ? U) a).
+
+ntheorem new_fish_antirefl:
+ ∀A:nAx.∀F:Ω^A.∀a. a ⋉ F → a ∈ F.
+#A; #F; #a; #H; nlapply (H (oO ?)); #aFo; napply aFo;
+nqed.
+
+nlemma co_ord_subset:
+ ∀A:nAx.∀F:Ω^A.∀a,i.∀f:𝐃 a i → Ord A.∀j. F⎽(Λ f) ⊆ F⎽(f j).
+#A; #F; #a; #i; #f; #j; #x; #H; napply H;
+nqed.
+
+naxiom AC_dual : 
+  ∀A:nAx.∀a:A.∀i,F. (∀f:𝐃 a i → Ord A.∃x:𝐃 a i.𝐝 a i x ∈ F⎽(f x)) → ∃j:𝐃 a i.∀x:Ord A.𝐝 a i j ∈ F⎽x.
+
+
+ntheorem new_fish_compatible: 
+ ∀A:nAx.∀F:Ω^A.∀a. a ⋉ F → ∀i:𝐈 a.∃j:𝐃 a i.𝐝 a i j ⋉ F.
+#A; #F; #a; #aF; #i; nnormalize;
+napply AC_dual; #f;
+nlapply (aF (Λf+1)); #aLf;
+nchange in aLf with (a ∈ F⎽(Λ f) ∧ ∀i:𝐈 a.∃j:𝐃 a i.𝐝 a i j ∈ F⎽(Λ f));
+ncut (∃j:𝐃 a i.𝐝 a i j ∈ F⎽(f j));##[
+  ncases aLf; #_; #H; nlapply (H i); *; #j; #Hj; @j; napply Hj;##] #aLf';
+napply aLf';
+nqed.
+
+(* move away *)
+nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
+#A; #U; #V; #W; *; #H; #x; *; #xU; #xV; napply H; nassumption;
+nqed.
+
+nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
+#A; #U; #V; #W; #H1; #H2; #x; #Hx; @; ##[ napply H1; ##| napply H2; ##] nassumption;
+nqed. 
+ 
+ntheorem max_new_fished: 
+  ∀A:nAx.∀G,F:Ω^A.G ⊆ F → (∀a.a ∈ G → ∀i.𝐈𝐦[𝐝 a i] ≬ G) → G ⊆ ⋉F.
+#A; #G; #F; #GF; #H; #b; #HbG; #o; ngeneralize in match HbG; ngeneralize in match b;
+nchange with (G ⊆ F⎽o);
+nelim o;
+##[ napply GF;
+##| #p; #IH; napply (subseteq_intersection_r … IH);
+    #x; #Hx; #i; ncases (H … Hx i); #c; *; *; #d; #Ed; #cG;
+    @d; napply IH; napply (setoidification … Ed^-1); napply cG;   
+##| #a; #i; #f; #Hf; nchange with (G ⊆ { y | ∀x. y ∈ F⎽(f x) }); 
+    #b; #Hb; #d; napply (Hf d); napply Hb;
 ##]
 nqed.
 
-STOP
-
-theorem reflexivity: ∀A:axiom_set.∀a:A.∀V. a ∈ V → a ◃ V.
- intros;
- apply refl;
- assumption.
-qed.
-
-theorem transitivity: ∀A:axiom_set.∀a:A.∀U,V. a ◃ U → U ◃ V → a ◃ V.
- intros;
- apply (covers_elim ?? {a | a ◃ V} ??? H); simplify; intros;
-  [ cases H1 in H2; apply H2;
-  | apply infinity;
-     [ assumption
-     | constructor 1;
-       assumption]]
-qed.
-
-theorem covers_elim2:
- ∀A: axiom_set. ∀U:Ω \sup A.∀P: A → CProp.
-  (∀a:A. a ∈ U → P a) →
-   (∀a:A.∀V:Ω \sup A. a ◃ V → V ◃ U → (∀y. y ∈ V → P y) → P a) →
-     ∀a:A. a ◃ U → P a.
- intros;
- change with (a ∈ {a | P a});
- apply (covers_elim ?????? H2);
-  [ intros 2; simplify; apply H; assumption
-  | intros;
-    simplify in H4 ⊢ %;
-    apply H1;
-     [ apply (C ? a1 j);
-     | autobatch; 
-     | assumption;
-     | assumption]]
-qed.
-
-theorem coreflexivity: ∀A:axiom_set.∀a:A.∀V. a ⋉ V → a ∈ V.
- intros;
- cases H;
- assumption.
-qed.
-
-theorem cotransitivity:
- ∀A:axiom_set.∀a:A.∀U,V. a ⋉ U → (∀b:A. b ⋉ U → b ∈ V) → a ⋉ V.
- intros;
- apply (fish_rec ?? {a|a ⋉ U} ??? H); simplify; intros;
-  [ apply H1; apply H2;
-  | cases H2 in j; clear H2; intro i;
-    cases (H4 i); clear H4; exists[apply a3] assumption]
-qed.
-
-theorem compatibility: ∀A:axiom_set.∀a:A.∀U,V. a ⋉ V → a ◃ U → U ⋉ V.
- intros;
- generalize in match H; clear H; 
- apply (covers_elim ?? {a|a ⋉ V → U ⋉ V} ??? H1);
- clear H1; simplify; intros;
-  [ exists [apply x] assumption
-  | cases H2 in j H H1; clear H2 a1; intros; 
-    cases (H1 i); clear H1; apply (H3 a1); assumption]
-qed.
-
-definition leq ≝ λA:axiom_set.λa,b:A. a ◃ {y|b=y}.
-
-interpretation "covered by one" 'leq a b = (leq ? a b).
-
-theorem leq_refl: ∀A:axiom_set.∀a:A. a ≤ a.
- intros;
- apply refl;
- normalize;
- reflexivity.
-qed.
-
-theorem leq_trans: ∀A:axiom_set.∀a,b,c:A. a ≤ b → b ≤ c → a ≤ c.
- intros;
- unfold in H H1 ⊢ %;
- apply (transitivity ???? H);
- constructor 1;
- intros;
- normalize in H2;
- rewrite < H2;
- assumption.
-qed.
-
-definition uparrow ≝ λA:axiom_set.λa:A.mk_powerset ? (λb:A. a ≤ b).
-
-interpretation "uparrow" 'uparrow a = (uparrow ? a).
-
-definition downarrow ≝ λA:axiom_set.λU:Ω \sup A.mk_powerset ? (λa:A. (↑a) ≬ U).
-
-interpretation "downarrow" 'downarrow a = (downarrow ? a).
-
-definition fintersects ≝ λA:axiom_set.λU,V:Ω \sup A.↓U ∩ ↓V.
-
-interpretation "fintersects" 'fintersects U V = (fintersects ? U V).
-
-record convergent_generated_topology : Type ≝
- { AA:> axiom_set;
-   convergence: ∀a:AA.∀U,V:Ω \sup AA. a ◃ U → a ◃ V → a ◃ (U ↓ V)
- }.