src/HOL/ex/set.ML
author paulson
Wed Nov 05 13:23:46 1997 +0100 (1997-11-05)
changeset 4153 e534c4c32d54
parent 4109 b131edcfeac3
child 4324 9bfac4684f2f
permissions -rw-r--r--
Ran expandshort, especially to introduce Safe_tac
     1 (*  Title:      HOL/ex/set.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
     4     Copyright   1991  University of Cambridge
     5 
     6 Cantor's Theorem; the Schroeder-Berstein Theorem.  
     7 *)
     8 
     9 
    10 writeln"File HOL/ex/set.";
    11 
    12 context Lfp.thy;
    13 
    14 (*Nice demonstration of blast_tac--and its limitations*)
    15 goal Set.thy "!!S::'a set set. ALL x:S. ALL y:S. x<=y ==> EX z. S <= {z}";
    16 (*for some unfathomable reason, UNIV_I increases the search space greatly*)
    17 by (blast_tac (claset() delrules [UNIV_I]) 1);
    18 result();
    19 
    20 
    21 (*** A unique fixpoint theorem --- fast/best/meson all fail ***)
    22 
    23 val [prem] = goal HOL.thy "?!x. f(g(x))=x ==> ?!y. g(f(y))=y";
    24 by (EVERY1[rtac (prem RS ex1E), rtac ex1I, etac arg_cong,
    25           rtac subst, atac, etac allE, rtac arg_cong, etac mp, etac arg_cong]);
    26 result();
    27 
    28 (*** Cantor's Theorem: There is no surjection from a set to its powerset. ***)
    29 
    30 goal Set.thy "~ (? f:: 'a=>'a set. ! S. ? x. f(x) = S)";
    31 (*requires best-first search because it is undirectional*)
    32 by (best_tac (claset() addSEs [equalityCE]) 1);
    33 qed "cantor1";
    34 
    35 (*This form displays the diagonal term*)
    36 goal Set.thy "! f:: 'a=>'a set. ! x. f(x) ~= ?S(f)";
    37 by (best_tac (claset() addSEs [equalityCE]) 1);
    38 uresult();
    39 
    40 (*This form exploits the set constructs*)
    41 goal Set.thy "?S ~: range(f :: 'a=>'a set)";
    42 by (rtac notI 1);
    43 by (etac rangeE 1);
    44 by (etac equalityCE 1);
    45 by (dtac CollectD 1);
    46 by (contr_tac 1);
    47 by (swap_res_tac [CollectI] 1);
    48 by (assume_tac 1);
    49 
    50 choplev 0;
    51 by (best_tac (claset() addSEs [equalityCE]) 1);
    52 
    53 (*** The Schroder-Berstein Theorem ***)
    54 
    55 goalw Lfp.thy [image_def] "!!f. inj(f) ==> inv(f)``(f``X) = X";
    56 by (rtac equalityI 1);
    57 by (fast_tac (claset() addEs [inv_f_f RS ssubst]) 1);
    58 by (fast_tac (claset() addEs [inv_f_f RS ssubst]) 1);
    59 qed "inv_image_comp";
    60 
    61 goal Set.thy "!!f. f(a) ~: (f``X) ==> a~:X";
    62 by (Blast_tac 1);
    63 qed "contra_imageI";
    64 
    65 goal Lfp.thy "(a ~: Compl(X)) = (a:X)";
    66 by (Blast_tac 1);
    67 qed "not_Compl";
    68 
    69 (*Lots of backtracking in this proof...*)
    70 val [compl,fg,Xa] = goal Lfp.thy
    71     "[| Compl(f``X) = g``Compl(X);  f(a)=g(b);  a:X |] ==> b:X";
    72 by (EVERY1 [rtac (not_Compl RS subst), rtac contra_imageI,
    73             rtac (compl RS subst), rtac (fg RS subst), stac not_Compl,
    74             rtac imageI, rtac Xa]);
    75 qed "disj_lemma";
    76 
    77 goalw Lfp.thy [image_def]
    78     "range(%z. if z:X then f(z) else g(z)) = f``X Un g``Compl(X)";
    79 by (simp_tac (simpset() addsplits [expand_if]) 1);
    80 by (Blast_tac 1);
    81 qed "range_if_then_else";
    82 
    83 goal Lfp.thy "a : X Un Compl(X)";
    84 by (Blast_tac 1);
    85 qed "X_Un_Compl";
    86 
    87 goalw Lfp.thy [surj_def] "surj(f) = (!a. a : range(f))";
    88 by (fast_tac (claset() addEs [ssubst]) 1);
    89 qed "surj_iff_full_range";
    90 
    91 val [compl] = goal Lfp.thy
    92     "Compl(f``X) = g``Compl(X) ==> surj(%z. if z:X then f(z) else g(z))";
    93 by (EVERY1[stac surj_iff_full_range, stac range_if_then_else,
    94            stac (compl RS sym)]);
    95 by (rtac (X_Un_Compl RS allI) 1);
    96 qed "surj_if_then_else";
    97 
    98 val [injf,injg,compl,bij] = goal Lfp.thy
    99     "[| inj_onto f X;  inj_onto g (Compl X);  Compl(f``X) = g``Compl(X); \
   100 \       bij = (%z. if z:X then f(z) else g(z)) |] ==> \
   101 \       inj(bij) & surj(bij)";
   102 val f_eq_gE = make_elim (compl RS disj_lemma);
   103 by (stac bij 1);
   104 by (rtac conjI 1);
   105 by (rtac (compl RS surj_if_then_else) 2);
   106 by (rewtac inj_def);
   107 by (cut_facts_tac [injf,injg] 1);
   108 by (EVERY1 [rtac allI, rtac allI, stac expand_if, rtac conjI, stac expand_if]);
   109 by (fast_tac (claset() addEs  [inj_ontoD, sym RS f_eq_gE]) 1);
   110 by (stac expand_if 1);
   111 by (fast_tac (claset() addEs  [inj_ontoD, f_eq_gE]) 1);
   112 qed "bij_if_then_else";
   113 
   114 goal Lfp.thy "? X. X = Compl(g``Compl((f:: 'a=>'b)``X))";
   115 by (rtac exI 1);
   116 by (rtac lfp_Tarski 1);
   117 by (REPEAT (ares_tac [monoI, image_mono, Compl_anti_mono] 1));
   118 qed "decomposition";
   119 
   120 val [injf,injg] = goal Lfp.thy
   121    "[| inj(f:: 'a=>'b);  inj(g:: 'b=>'a) |] ==> \
   122 \   ? h:: 'a=>'b. inj(h) & surj(h)";
   123 by (rtac (decomposition RS exE) 1);
   124 by (rtac exI 1);
   125 by (rtac bij_if_then_else 1);
   126 by (EVERY [rtac refl 4, rtac (injf RS inj_imp) 1,
   127            rtac (injg RS inj_onto_inv) 1]);
   128 by (EVERY1 [etac ssubst, stac double_complement, rtac subsetI,
   129             etac imageE, etac ssubst, rtac rangeI]);
   130 by (EVERY1 [etac ssubst, stac double_complement, 
   131             rtac (injg RS inv_image_comp RS sym)]);
   132 qed "schroeder_bernstein";
   133 
   134 writeln"Reached end of file.";