Not is now inductive.

[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 f9655f4ca8f86dbcb36af2a5069679b193310ae9..d3dccb8ee07e66248e17e69f21536cc6fb3013fe 100644 (file)
--- a/helm/software/matita/nlibrary/topology/cantor.ma
+++ b/helm/software/matita/nlibrary/topology/cantor.ma
@@ -1,4 +1,16 @@
-
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
 
 include "topology/igft.ma".
 
@@ -7,14 +19,14 @@ ntheorem axiom_cond: ∀A:Ax.∀a:A.∀i:𝐈 a.a ◃ 𝐂 a i.
 nqed.
 
 nlemma hint_auto1 : ∀A,U,V. (∀x.x ∈ U → x ◃ V) → cover_set cover A U V.
-nnormalize; nauto.
+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; nauto depth=4;
+#A; #a; #U; #V; #aU; #UV; nelim aU; /3/;
 nqed.
 
 ndefinition emptyset: ∀A.Ω^A ≝ λA.{x | False}.
@@ -29,7 +41,7 @@ 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); nauto;
+##| #b; #i_star; #IH1; #IH2; ncases (EM … b i_star); /3/;
 ##] 
 nqed.
 
@@ -50,7 +62,7 @@ nrecord uAx : Type[1] ≝ {
 
 ndefinition uax : uAx → Ax.
 #A; @ (uax_ A) (λx.unit); #a; #_; 
-napply (𝐂 a ?);  nlapply one; ncases (with_ A a); nauto; 
+napply (𝐂 a ?);  nlapply one; ncases (with_ A a); //; 
 nqed.
 
 ncoercion uax : ∀u:uAx. Ax ≝ uax on _u : uAx to Ax.
@@ -68,30 +80,29 @@ unification hint 0 ≔ ;
   (* -------------- *) ⊢
          S x ≡ A. 
 
-
 ntheorem col2_4 :
-  ∀A:uAx.∀a:A. a ◃ ∅ → ¬ a ∈ 𝐂 a one. 
+  ∀A:uAx.∀a:uax A. a ◃ ∅ → ¬ a ∈ 𝐂 a one.
 #A; #a; #H; nelim H;
 ##[ #n; *;
-##| #b; #i_star; #IH1; #IH2; #H3; nlapply (IH2 … H3); nauto;
-##] 
+##| #b; #i_star; #IH1; #IH2; #H3; nlapply (IH2 … H3); /2/;
+##]
 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; nauto depth=4; 
+#A; #a; #U; #V; #HUV; #H; nelim H; /3/; 
 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);
-  (* nauto; *) napply (cover_monotone … AxCon); nassumption; ##] #H; 
-ncut (a ◃ ∅); ##[ napply (transitivity … H); #x; #E; napply E; ##] #H1;
+  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;
-nauto;
+/2/;
 nqed.
 
 include "nat/nat.ma".
@@ -127,22 +138,18 @@ 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; nauto;
+#A; #U; #V;  #UV; #VU; #a; #aU; nelim aU; /3/;
 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);
-     (* nauto; *) #H1; #H2; #H3; nauto; (* ??? *) ##]
-  napply (replace_char … { x | ϕ a x = O }); ##[##1,2: #x; nrewrite > (H x);
-     (* nauto; *) #E; napply E; ##]
+  napply (replace_char … { x | h x = O }); ##[ ##1,2: #x; ncases (Ph x); /2/; ##]
+  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); #x; nauto; ##] #H2;
+ncut (a ◃ ∅); ##[ napply (transitivity … H1); //; ##] #H2;
 nlapply (col2_4 …H2); #H3;
 ncut (a ∈ 𝐂 a one); ##[
-  nnormalize; ncases (Ph a); nrewrite > (H a); nauto; ##] #H4;
-nauto;
-nqed.
-
-
+  nnormalize; ncases (Ph a); nrewrite > (H a); /2/; ##] #H4;
+/2/;
+nqed.
\ No newline at end of file