--- a/ex/set.ML Tue Oct 24 14:59:17 1995 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,132 +0,0 @@
-(* 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.";
-
-(*** A unique fixpoint theorem --- fast/best/meson all fail ***)
-
-val [prem] = goal HOL.thy "?!x.f(g(x))=x ==> ?!y.g(f(y))=y";
-by(EVERY1[rtac (prem RS ex1E), rtac ex1I, etac arg_cong,
- rtac subst, atac, etac allE, rtac arg_cong, etac mp, etac arg_cong]);
-result();
-
-(*** 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);
-qed "cantor1";
-
-(*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);
-qed "inv_image_comp";
-
-val prems = goal Set.thy "f(a) ~: (f``X) ==> a~:X";
-by (cfast_tac prems 1);
-qed "contra_imageI";
-
-goal Lfp.thy "(a ~: Compl(X)) = (a:X)";
-by (fast_tac set_cs 1);
-qed "not_Compl";
-
-(*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]);
-qed "disj_lemma";
-
-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);
-qed "range_if_then_else";
-
-goal Lfp.thy "a : X Un Compl(X)";
-by (fast_tac set_cs 1);
-qed "X_Un_Compl";
-
-goalw Lfp.thy [surj_def] "surj(f) = (!a. a : range(f))";
-by (fast_tac (set_cs addEs [ssubst]) 1);
-qed "surj_iff_full_range";
-
-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);
-qed "surj_if_then_else";
-
-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);
-qed "bij_if_then_else";
-
-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));
-qed "decomposition";
-
-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)]);
-qed "schroeder_bernstein";
-
-writeln"Reached end of file.";