// in place of nauto everywhere

[helm.git] / helm / software / matita / nlibrary / topology / cantor.ma

diff --git a/helm/software/matita/nlibrary/topology/cantor.ma b/helm/software/matita/nlibrary/topology/cantor.ma
index 95cf6b3aaf4007d7cc3d096afa43f7ea4a6976d1..e7f9a5a15bed6ccbf0709169558c16a2497ff0dc 100644 (file)
--- a/helm/software/matita/nlibrary/topology/cantor.ma
+++ b/helm/software/matita/nlibrary/topology/cantor.ma
@@ -2,45 +2,145 @@
 
 include "topology/igft.ma".
 
+ntheorem axiom_cond: ∀A:Ax.∀a:A.∀i:𝐈 a.a ◃ 𝐂 a i.
+#A; #a; #i; @2 i; #x; #H; @; napply H;
+nqed.
+
+nlemma hint_auto1 : ∀A,U,V. (∀x.x ∈ U → x ◃ V) → cover_set cover A U V.
+nnormalize; /2/.
+nqed.
+
+alias symbol "covers" (instance 1) = "covers".
+alias symbol "covers" (instance 2) = "covers set".
+alias symbol "covers" (instance 3) = "covers".
+ntheorem transitivity: ∀A:Ax.∀a:A.∀U,V. a ◃ U → U ◃ V → a ◃ V.
+#A; #a; #U; #V; #aU; #UV; nelim aU; /3/;
+nqed.
+
+ndefinition emptyset: ∀A.Ω^A ≝ λA.{x | False}.
+
+notation "∅" non associative with precedence 90 for @{ 'empty }.
+interpretation "empty" 'empty = (emptyset ?).
+
+naxiom EM : ∀A:Ax.∀a:A.∀i_star.(a ∈ 𝐂 a i_star) ∨ ¬( a ∈ 𝐂 a i_star).
+
+alias symbol "covers" = "covers".
+ntheorem th2_3 :
+  ∀A:Ax.∀a:A. a ◃ ∅ → ∃i. ¬ a ∈ 𝐂 a i.
+#A; #a; #H; nelim H;
+##[ #n; *;
+##| #b; #i_star; #IH1; #IH2; ncases (EM … b i_star); /2/;
+##] 
+nqed.
+
+ninductive eq1 (A : Type[0]) : Type[0] → CProp[0] ≝ 
+| refl1 : eq1 A A.
+
+notation "hvbox( a break ∼ b)" non associative with precedence 40 
+for @{ 'eqT $a $b }.
+
+interpretation "eq between types" 'eqT a b = (eq1 a b).
+
 ninductive unit : Type[0] ≝ one : unit.
 
-naxiom E: setoid.
-naxiom R: E → Ω^E.
+nrecord uAx : Type[1] ≝ {
+  uax_ : Ax;
+  with_ : ∀a:uax_.𝐈 a ∼ unit
+}.
 
-ndefinition axs: Ax.
-@ E (λ_.unit) (λa,x.R a);
+ndefinition uax : uAx → Ax.
+#A; @ (uax_ A) (λx.unit); #a; #_; 
+napply (𝐂 a ?);  nlapply one; ncases (with_ A a); //; 
 nqed.
 
+ncoercion uax : ∀u:uAx. Ax ≝ uax on _u : uAx to Ax.
+
+naxiom A: Type[0].
+naxiom S: A → Ω^A.
+
+ndefinition axs: uAx.
+@; ##[ @ A (λ_.unit) (λa,x.S a); ##| #_; @; ##]
+nqed.
+ 
+alias id "S" = "cic:/matita/ng/topology/igft/S.fix(0,0,1)".
 unification hint 0 ≔ ;
          x ≟ axs  
   (* -------------- *) ⊢
-         S x ≡ E. 
-         
-ndefinition emptyset: Ω^axs ≝ {x | False}.
+         S x ≡ A. 
 
-ndefinition Z: Ω^axs ≝ {x | x ◃ emptyset}.
 
+ntheorem col2_4 :
+  ∀A:uAx.∀a:A. a ◃ ∅ → ¬ a ∈ 𝐂 a one. 
+#A; #a; #H; nelim H;
+##[ #n; *;
+##| #b; #i_star; #IH1; #IH2; #H3; nlapply (IH2 … H3); //;
+##] 
+nqed.
+
+ndefinition Z : Ω^axs ≝ { x | x ◃ ∅ }.
+
+ntheorem cover_monotone: ∀A:Ax.∀a:A.∀U,V.U ⊆ V → a ◃ U → a ◃ V.
+#A; #a; #U; #V; #HUV; #H; nelim H; //; 
+nqed.
+
+ntheorem th3_1: ¬∃a:axs.Z ⊆ S a ∧ S a ⊆ Z.
+*; #a; *; #ZSa; #SaZ; 
+ncut (a ◃ Z); ##[
+  nlapply (axiom_cond … a one); #AxCon; nchange in AxCon with (a ◃ S a);
+  napply (cover_monotone … AxCon); nassumption; ##] #H; 
+ncut (a ◃ ∅); ##[ napply (transitivity … H); nwhd in match Z; //; ##] #H1;
+ncut (¬ a ∈ S a); ##[ napply (col2_4 … H1); ##] #H2;
+ncut (a ∈ S a); ##[ napply ZSa; napply H1; ##] #H3;
+//;
+nqed.
+
+include "nat/nat.ma".
+
+naxiom phi : nat → nat → nat.
+
+notation > "ϕ" non associative with precedence 90 for @{ 'phi }.
+interpretation "phi" 'phi = phi.
+ 
+notation < "ϕ a i" non associative with precedence 90 for @{ 'phi2 $a $i}.
+interpretation "phi2" 'phi2 a i = (phi a i). 
+notation < "ϕ a" non associative with precedence 90 for @{ 'phi1 $a }.
+interpretation "phi2" 'phi1 a = (phi a). 
+
+ndefinition caxs : uAx. 
+@; ##[ @ nat (λ_.unit); #a; #_; napply { x | ϕ a x = O } ##| #_; @; ##]
+nqed.
+
+
+alias id "S" = "cic:/matita/ng/topology/igft/S.fix(0,0,1)".
+unification hint 0 ≔ ;
+         x ≟ caxs  
+  (* -------------- *) ⊢
+         S x ≡ nat. 
+
+naxiom h : nat → nat. 
+
+alias symbol "eq" = "leibnitz's equality".
+alias symbol "eq" = "setoid1 eq".
 alias symbol "covers" = "covers".
-alias symbol "covers" = "covers set".
-alias symbol "covers" = "covers".
-alias symbol "covers" = "covers set".
-ntheorem cover_trans: ∀A:Ax.∀a:A.∀U,V. a ◃ U → U ◃ V → a ◃ V.
-#A; #a; #U; #V; #aU; #UV;
-nelim aU;
-##[ #c; #H; napply (UV … H);
-##| #c; #i; #HCU; #H; napply (cinfinity … i); napply H;
-##]
-nqed.
-
-ntheorem cantor: ∀a:axs. ¬ (Z ⊆ R a ∧ R a ⊆ Z).
-#a; *; #ZRa; #RaZ;
-ncut (a ◃ R a); ##[ @2; ##[ napply one; ##] #x; #H; @; napply H; ##] #H1;
-ncut (a ◃ emptyset); ##[
-  napply (cover_trans … H1); 
-  #x; #H; nlapply (RaZ … H); #ABS; napply ABS; ##] #H2;
-ncut (a ∈ R a); ##[ napply ZRa; napply H2; ##] #H3;
-nelim H2 in H3; 
-##[ nauto.
-##| nnormalize; nauto. ##] (* se lo lancio su entrambi fallisce di width *)
+alias symbol "eq" = "leibnitz's equality".
+naxiom Ph :  ∀x.h x = O \liff  x ◃ ∅.
+
+nlemma replace_char:
+  ∀A:Ax.∀U,V.U ⊆ V → V ⊆ U → ∀a:A.a ◃ U → a ◃ V.
+#A; #U; #V;  #UV; #VU; #a; #aU; nelim aU; //;
 nqed.
 
+ntheorem th_ch3: ¬∃a:caxs.∀x.ϕ a x = h x.
+*; #a; #H;
+ncut (a ◃ { x | x ◃ ∅}); ##[
+  napply (replace_char … { x | h x = O }); ##[ ##1,2: #x; ncases (Ph x); //; ##]
+  napply (replace_char … { x | ϕ a x = O }); ##[##1,2: #x; nrewrite > (H x); //; ##]
+  napply (axiom_cond … a one); ##] #H1;
+ncut (a ◃ ∅); ##[ napply (transitivity … H1); //; ##] #H2;
+nlapply (col2_4 …H2); #H3;
+ncut (a ∈ 𝐂 a one); ##[
+  nnormalize; ncases (Ph a); nrewrite > (H a); //; ##] #H4;
+//;
+nqed.
+
+