ex/set.ML

used ~= for "not equals" and ~: for "not in"

(*  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.";