clasohm@1465: (*  Title:      HOL/ex/set.ML
clasohm@969:     ID:         $Id$
clasohm@1465:     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
clasohm@969:     Copyright   1991  University of Cambridge
clasohm@969: 
clasohm@969: Cantor's Theorem; the Schroeder-Berstein Theorem.  
clasohm@969: *)
clasohm@969: 
clasohm@969: 
clasohm@969: writeln"File HOL/ex/set.";
clasohm@969: 
paulson@4153: context Lfp.thy;
paulson@2998: 
paulson@4153: (*Nice demonstration of blast_tac--and its limitations*)
paulson@2998: goal Set.thy "!!S::'a set set. ALL x:S. ALL y:S. x<=y ==> EX z. S <= {z}";
paulson@4153: (*for some unfathomable reason, UNIV_I increases the search space greatly*)
paulson@4153: by (blast_tac (claset() delrules [UNIV_I]) 1);
paulson@4153: result();
paulson@4153: 
paulson@2998: 
clasohm@969: (*** A unique fixpoint theorem --- fast/best/meson all fail ***)
clasohm@969: 
wenzelm@3842: val [prem] = goal HOL.thy "?!x. f(g(x))=x ==> ?!y. g(f(y))=y";
paulson@2031: by (EVERY1[rtac (prem RS ex1E), rtac ex1I, etac arg_cong,
clasohm@969:           rtac subst, atac, etac allE, rtac arg_cong, etac mp, etac arg_cong]);
clasohm@969: result();
clasohm@969: 
clasohm@969: (*** Cantor's Theorem: There is no surjection from a set to its powerset. ***)
clasohm@969: 
clasohm@969: goal Set.thy "~ (? f:: 'a=>'a set. ! S. ? x. f(x) = S)";
clasohm@969: (*requires best-first search because it is undirectional*)
wenzelm@4089: by (best_tac (claset() addSEs [equalityCE]) 1);
clasohm@969: qed "cantor1";
clasohm@969: 
clasohm@969: (*This form displays the diagonal term*)
clasohm@969: goal Set.thy "! f:: 'a=>'a set. ! x. f(x) ~= ?S(f)";
wenzelm@4089: by (best_tac (claset() addSEs [equalityCE]) 1);
clasohm@969: uresult();
clasohm@969: 
clasohm@969: (*This form exploits the set constructs*)
clasohm@969: goal Set.thy "?S ~: range(f :: 'a=>'a set)";
clasohm@969: by (rtac notI 1);
clasohm@969: by (etac rangeE 1);
clasohm@969: by (etac equalityCE 1);
clasohm@969: by (dtac CollectD 1);
clasohm@969: by (contr_tac 1);
clasohm@969: by (swap_res_tac [CollectI] 1);
clasohm@969: by (assume_tac 1);
clasohm@969: 
clasohm@969: choplev 0;
wenzelm@4089: by (best_tac (claset() addSEs [equalityCE]) 1);
clasohm@969: 
clasohm@969: (*** The Schroder-Berstein Theorem ***)
clasohm@969: 
paulson@2935: goalw Lfp.thy [image_def] "!!f. inj(f) ==> inv(f)``(f``X) = X";
clasohm@969: by (rtac equalityI 1);
wenzelm@4089: by (fast_tac (claset() addEs [inv_f_f RS ssubst]) 1);
wenzelm@4089: by (fast_tac (claset() addEs [inv_f_f RS ssubst]) 1);
clasohm@969: qed "inv_image_comp";
clasohm@969: 
paulson@1888: goal Set.thy "!!f. f(a) ~: (f``X) ==> a~:X";
paulson@2935: by (Blast_tac 1);
clasohm@969: qed "contra_imageI";
clasohm@969: 
clasohm@969: goal Lfp.thy "(a ~: Compl(X)) = (a:X)";
paulson@2935: by (Blast_tac 1);
clasohm@969: qed "not_Compl";
clasohm@969: 
clasohm@969: (*Lots of backtracking in this proof...*)
clasohm@969: val [compl,fg,Xa] = goal Lfp.thy
clasohm@969:     "[| Compl(f``X) = g``Compl(X);  f(a)=g(b);  a:X |] ==> b:X";
clasohm@969: by (EVERY1 [rtac (not_Compl RS subst), rtac contra_imageI,
clasohm@1465:             rtac (compl RS subst), rtac (fg RS subst), stac not_Compl,
clasohm@1465:             rtac imageI, rtac Xa]);
clasohm@969: qed "disj_lemma";
clasohm@969: 
paulson@1888: goalw Lfp.thy [image_def]
paulson@1888:     "range(%z. if z:X then f(z) else g(z)) = f``X Un g``Compl(X)";
wenzelm@4089: by (simp_tac (simpset() addsplits [expand_if]) 1);
paulson@2935: by (Blast_tac 1);
clasohm@969: qed "range_if_then_else";
clasohm@969: 
clasohm@969: goal Lfp.thy "a : X Un Compl(X)";
paulson@2935: by (Blast_tac 1);
clasohm@969: qed "X_Un_Compl";
clasohm@969: 
clasohm@969: goalw Lfp.thy [surj_def] "surj(f) = (!a. a : range(f))";
wenzelm@4089: by (fast_tac (claset() addEs [ssubst]) 1);
clasohm@969: qed "surj_iff_full_range";
clasohm@969: 
clasohm@969: val [compl] = goal Lfp.thy
clasohm@969:     "Compl(f``X) = g``Compl(X) ==> surj(%z. if z:X then f(z) else g(z))";
nipkow@1487: by (EVERY1[stac surj_iff_full_range, stac range_if_then_else,
nipkow@1487:            stac (compl RS sym)]);
clasohm@969: by (rtac (X_Un_Compl RS allI) 1);
clasohm@969: qed "surj_if_then_else";
clasohm@969: 
clasohm@969: val [injf,injg,compl,bij] = goal Lfp.thy
clasohm@969:     "[| inj_onto f X;  inj_onto g (Compl X);  Compl(f``X) = g``Compl(X); \
clasohm@969: \       bij = (%z. if z:X then f(z) else g(z)) |] ==> \
clasohm@969: \       inj(bij) & surj(bij)";
clasohm@969: val f_eq_gE = make_elim (compl RS disj_lemma);
paulson@2031: by (stac bij 1);
clasohm@969: by (rtac conjI 1);
clasohm@969: by (rtac (compl RS surj_if_then_else) 2);
clasohm@969: by (rewtac inj_def);
clasohm@969: by (cut_facts_tac [injf,injg] 1);
clasohm@969: by (EVERY1 [rtac allI, rtac allI, stac expand_if, rtac conjI, stac expand_if]);
wenzelm@4089: by (fast_tac (claset() addEs  [inj_ontoD, sym RS f_eq_gE]) 1);
clasohm@969: by (stac expand_if 1);
wenzelm@4089: by (fast_tac (claset() addEs  [inj_ontoD, f_eq_gE]) 1);
clasohm@969: qed "bij_if_then_else";
clasohm@969: 
clasohm@969: goal Lfp.thy "? X. X = Compl(g``Compl((f:: 'a=>'b)``X))";
clasohm@969: by (rtac exI 1);
clasohm@969: by (rtac lfp_Tarski 1);
clasohm@969: by (REPEAT (ares_tac [monoI, image_mono, Compl_anti_mono] 1));
clasohm@969: qed "decomposition";
clasohm@969: 
clasohm@969: val [injf,injg] = goal Lfp.thy
clasohm@969:    "[| inj(f:: 'a=>'b);  inj(g:: 'b=>'a) |] ==> \
clasohm@969: \   ? h:: 'a=>'b. inj(h) & surj(h)";
clasohm@969: by (rtac (decomposition RS exE) 1);
clasohm@969: by (rtac exI 1);
clasohm@969: by (rtac bij_if_then_else 1);
clasohm@969: by (EVERY [rtac refl 4, rtac (injf RS inj_imp) 1,
nipkow@2911:            rtac (injg RS inj_onto_inv) 1]);
clasohm@969: by (EVERY1 [etac ssubst, stac double_complement, rtac subsetI,
clasohm@1465:             etac imageE, etac ssubst, rtac rangeI]);
clasohm@969: by (EVERY1 [etac ssubst, stac double_complement, 
clasohm@1465:             rtac (injg RS inv_image_comp RS sym)]);
clasohm@969: qed "schroeder_bernstein";
clasohm@969: 
clasohm@969: writeln"Reached end of file.";