-
+(**************************************************************************)
+(* ___ *)
+(* ||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".
nqed.
nlemma hint_auto1 : âA,U,V. (âx.x â U â x â V) â 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: âA:Ax.âa:A.âU,V. a â U â U â V â a â V.
-#A; #a; #U; #V; #aU; #UV; nelim aU; nauto depth=4;
+#A; #a; #U; #V; #aU; #UV; nelim aU; /3/;
nqed.
ndefinition emptyset: âA.Ί^A â ÎťA.{x | False}.
âA:Ax.âa:A. a â â
â âi. ÂŹ a â đ 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.
ndefinition uax : uAx â Ax.
#A; @ (uax_ A) (Îťx.unit); #a; #_;
-napply (đ a ?); nlapply one; ncases (with_ A a); nauto;
+napply (đ a ?); nlapply one; ncases (with_ A a); //;
nqed.
ncoercion uax : âu:uAx. Ax â uax on _u : uAx to Ax.
unification hint 0 â ;
x â axs
(* -------------- *) â˘
- S x ⥠A.
-
+ S (uax x) ⥠A. (* XXX: bug coercions/ disamb multipasso che ne fa 1 solo*)
ntheorem col2_4 :
- âA:uAx.âa:A. a â â
â ÂŹ a â đ a one.
+ âA:uAx.âa:uax A. a â â
â ÂŹ a â đ 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 : Ί^axs â { x | x â â
}.
+(* bug interpretazione non aggiunta per â
*)
+ndefinition Z : Ί^axs â { x | x â (emptyset ?) }.
ntheorem cover_monotone: âA:Ax.âa:A.âU,V.U â V â a â U â a â 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: ÂŹâa:axs.Z â S a ⧠S a â Z.
*; #a; *; #ZSa; #SaZ;
ncut (a â Z); ##[
nlapply (axiom_cond ⌠a one); #AxCon; nchange in AxCon with (a â S a);
- (* nauto; *) napply (cover_monotone ⌠AxCon); nassumption; ##] #H;
-ncut (a â â
); ##[ napply (transitivity ⌠H); #x; #E; napply E; ##] #H1;
+ napply (cover_monotone ⌠AxCon); nassumption; ##] #H;
+ncut (a â â
); ##[ napply (transitivity ⌠H); nwhd in match Z; //; ##] #H1;
ncut (ÂŹ a â S a); ##[ napply (col2_4 ⌠H1); ##] #H2;
ncut (a â S a); ##[ napply ZSa; napply H1; ##] #H3;
-nauto;
+/2/;
nqed.
include "nat/nat.ma".
unification hint 0 â ;
x â caxs
(* -------------- *) â˘
- S x ⥠nat.
+ S (uax x) ⥠nat.
naxiom h : nat â nat.
nlemma replace_char:
âA:Ax.âU,V.U â V â V â U â âa:A.a â U â a â 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: ÂŹâa:caxs.âx.Ď a x = h x.
*; #a; #H;
ncut (a â { x | x â â
}); ##[
- napply (replace_char ⌠{ x | h x = O }); ##[ ##1,2: #x; ncases (Ph x); nauto; ##]
- napply (replace_char ⌠{ x | Ď a x = O }); ##[##1,2: #x; nrewrite > (H x); nauto; ##]
+ napply (replace_char ⌠{ x | h x = O }); ##[ ##1,2: #x; ncases (Ph x); /2/; ##]
+ napply (replace_char ⌠{ x | Ď a x = O }); ##[##1,2: #x; nrewrite > (H x); //; ##]
napply (axiom_cond ⌠a one); ##] #H1;
-ncut (a â â
); ##[ napply (transitivity ⌠H1); nauto; ##] #H2;
+ncut (a â â
); ##[ napply (transitivity ⌠H1); //; ##] #H2;
nlapply (col2_4 âŚH2); #H3;
ncut (a â đ 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