ex/set.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/ex/set.ML	Thu Sep 16 12:21:07 1993 +0200
@@ -0,0 +1,125 @@
+(*  Title: 	HOL/ex/set.ML
+    ID:         $Id$
+    Author: 	Tobias Nipkow, Cambridge University Computer Laboratory
+    Copyright   1991  University of Cambridge
+
+Cantor's Theorem; the Schroeder-Berstein Theorem.  
+*)
+
+
+writeln"File HOL/ex/set.";
+
+(*** Cantor's Theorem: There is no surjection from a set to its powerset. ***)
+
+goal Set.thy "~ (? f:: 'a=>'a set. ! S. ? x. f(x) = S)";
+(*requires best-first search because it is undirectional*)
+by (best_tac (set_cs addSEs [equalityCE]) 1);
+val cantor1 = result();
+
+(*This form displays the diagonal term*)
+goal Set.thy "! f:: 'a=>'a set. ! x. ~ f(x) = ?S(f)";
+by (best_tac (set_cs addSEs [equalityCE]) 1);
+uresult();
+
+(*This form exploits the set constructs*)
+goal Set.thy "~ ?S : range(f :: 'a=>'a set)";
+by (rtac notI 1);
+by (etac rangeE 1);
+by (etac equalityCE 1);
+by (dtac CollectD 1);
+by (contr_tac 1);
+by (swap_res_tac [CollectI] 1);
+by (assume_tac 1);
+
+choplev 0;
+by (best_tac (set_cs addSEs [equalityCE]) 1);
+
+(*** The Schroder-Berstein Theorem ***)
+
+val prems = goalw Lfp.thy [image_def] "inj(f) ==> Inv(f)``(f``X) = X";
+by (cut_facts_tac prems 1);
+by (rtac equalityI 1);
+by (fast_tac (set_cs addEs [Inv_f_f RS ssubst]) 1);
+by (fast_tac (set_cs addEs [Inv_f_f RS ssubst]) 1);
+val inv_image_comp = result();
+
+val prems = goal Set.thy "~ f(a) : (f``X) ==> ~ a:X";
+by (cfast_tac prems 1);
+val contra_imageI = result();
+
+goal Lfp.thy "(~ a : Compl(X)) = (a:X)";
+by (fast_tac set_cs 1);
+val not_Compl = result();
+
+(*Lots of backtracking in this proof...*)
+val [compl,fg,Xa] = goal Lfp.thy
+    "[| Compl(f``X) = g``Compl(X);  f(a)=g(b);  a:X |] ==> b:X";
+by (EVERY1 [rtac (not_Compl RS subst), rtac contra_imageI,
+	    rtac (compl RS subst), rtac (fg RS subst), stac not_Compl,
+	    rtac imageI, rtac Xa]);
+val disj_lemma = result();
+
+goal Lfp.thy "range(%z. if(z:X, f(z), g(z))) = f``X Un g``Compl(X)";
+by (rtac equalityI 1);
+by (rewtac range_def);
+by (fast_tac (set_cs addIs [if_P RS sym, if_not_P RS sym]) 2);
+by (rtac subsetI 1);
+by (etac CollectE 1);
+by (etac exE 1);
+by (etac ssubst 1);
+by (rtac (excluded_middle RS disjE) 1);
+by (EVERY' [rtac (if_P     RS ssubst), atac, fast_tac set_cs] 2);
+by (EVERY' [rtac (if_not_P RS ssubst), atac, fast_tac set_cs] 1);
+val range_if_then_else = result();
+
+goal Lfp.thy "a : X Un Compl(X)";
+by (fast_tac set_cs 1);
+val X_Un_Compl = result();
+
+goalw Lfp.thy [surj_def] "surj(f) = (!a. a : range(f))";
+by (fast_tac (set_cs addEs [ssubst]) 1);
+val surj_iff_full_range = result();
+
+val [compl] = goal Lfp.thy
+    "Compl(f``X) = g``Compl(X) ==> surj(%z. if(z:X, f(z), g(z)))";
+by (sstac [surj_iff_full_range, range_if_then_else, compl RS sym] 1);
+by (rtac (X_Un_Compl RS allI) 1);
+val surj_if_then_else = result();
+
+val [injf,injg,compl,bij] = goal Lfp.thy
+    "[| inj_onto(f,X);  inj_onto(g,Compl(X));  Compl(f``X) = g``Compl(X); \
+\       bij = (%z. if(z:X, f(z), g(z))) |] ==> \
+\       inj(bij) & surj(bij)";
+val f_eq_gE = make_elim (compl RS disj_lemma);
+by (rtac (bij RS ssubst) 1);
+by (rtac conjI 1);
+by (rtac (compl RS surj_if_then_else) 2);
+by (rewtac inj_def);
+by (cut_facts_tac [injf,injg] 1);
+by (EVERY1 [rtac allI, rtac allI, stac expand_if, rtac conjI, stac expand_if]);
+by (fast_tac (set_cs addEs  [inj_ontoD, sym RS f_eq_gE]) 1);
+by (stac expand_if 1);
+by (fast_tac (set_cs addEs  [inj_ontoD, f_eq_gE]) 1);
+val bij_if_then_else = result();
+
+goal Lfp.thy "? X. X = Compl(g``Compl(f:: 'a=>'b``X))";
+by (rtac exI 1);
+by (rtac lfp_Tarski 1);
+by (REPEAT (ares_tac [monoI, image_mono, Compl_anti_mono] 1));
+val decomposition = result();
+
+val [injf,injg] = goal Lfp.thy
+   "[| inj(f:: 'a=>'b);  inj(g:: 'b=>'a) |] ==> \
+\   ? h:: 'a=>'b. inj(h) & surj(h)";
+by (rtac (decomposition RS exE) 1);
+by (rtac exI 1);
+by (rtac bij_if_then_else 1);
+by (EVERY [rtac refl 4, rtac (injf RS inj_imp) 1,
+	   rtac (injg RS inj_onto_Inv) 1]);
+by (EVERY1 [etac ssubst, stac double_complement, rtac subsetI,
+	    etac imageE, etac ssubst, rtac rangeI]);
+by (EVERY1 [etac ssubst, stac double_complement, 
+	    rtac (injg RS inv_image_comp RS sym)]);
+val schroeder_bernstein = result();
+
+writeln"Reached end of file.";