src/HOL/ex/Set_Theory.thy
author wenzelm
Wed Jun 22 10:09:20 2016 +0200 (2016-06-22)
changeset 63343 fb5d8a50c641
parent 61945 1135b8de26c3
child 63804 70554522bf98
permissions -rw-r--r--
bundle lifting_syntax;
     1 (*  Title:      HOL/ex/Set_Theory.thy
     2     Author:     Tobias Nipkow and Lawrence C Paulson
     3     Copyright   1991  University of Cambridge
     4 *)
     5 
     6 section \<open>Set Theory examples: Cantor's Theorem, Schröder-Bernstein Theorem, etc.\<close>
     7 
     8 theory Set_Theory
     9 imports Main
    10 begin
    11 
    12 text\<open>
    13   These two are cited in Benzmueller and Kohlhase's system description
    14   of LEO, CADE-15, 1998 (pages 139-143) as theorems LEO could not
    15   prove.
    16 \<close>
    17 
    18 lemma "(X = Y \<union> Z) =
    19     (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"
    20   by blast
    21 
    22 lemma "(X = Y \<inter> Z) =
    23     (X \<subseteq> Y \<and> X \<subseteq> Z \<and> (\<forall>V. V \<subseteq> Y \<and> V \<subseteq> Z \<longrightarrow> V \<subseteq> X))"
    24   by blast
    25 
    26 text \<open>
    27   Trivial example of term synthesis: apparently hard for some provers!
    28 \<close>
    29 
    30 schematic_goal "a \<noteq> b \<Longrightarrow> a \<in> ?X \<and> b \<notin> ?X"
    31   by blast
    32 
    33 
    34 subsection \<open>Examples for the \<open>blast\<close> paper\<close>
    35 
    36 lemma "(\<Union>x \<in> C. f x \<union> g x) = \<Union>(f ` C)  \<union>  \<Union>(g ` C)"
    37   \<comment> \<open>Union-image, called \<open>Un_Union_image\<close> in Main HOL\<close>
    38   by blast
    39 
    40 lemma "(\<Inter>x \<in> C. f x \<inter> g x) = \<Inter>(f ` C) \<inter> \<Inter>(g ` C)"
    41   \<comment> \<open>Inter-image, called \<open>Int_Inter_image\<close> in Main HOL\<close>
    42   by blast
    43 
    44 lemma singleton_example_1:
    45      "\<And>S::'a set set. \<forall>x \<in> S. \<forall>y \<in> S. x \<subseteq> y \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
    46   by blast
    47 
    48 lemma singleton_example_2:
    49      "\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
    50   \<comment> \<open>Variant of the problem above.\<close>
    51   by blast
    52 
    53 lemma "\<exists>!x. f (g x) = x \<Longrightarrow> \<exists>!y. g (f y) = y"
    54   \<comment> \<open>A unique fixpoint theorem --- \<open>fast\<close>/\<open>best\<close>/\<open>meson\<close> all fail.\<close>
    55   by metis
    56 
    57 
    58 subsection \<open>Cantor's Theorem: There is no surjection from a set to its powerset\<close>
    59 
    60 lemma cantor1: "\<not> (\<exists>f:: 'a \<Rightarrow> 'a set. \<forall>S. \<exists>x. f x = S)"
    61   \<comment> \<open>Requires best-first search because it is undirectional.\<close>
    62   by best
    63 
    64 schematic_goal "\<forall>f:: 'a \<Rightarrow> 'a set. \<forall>x. f x \<noteq> ?S f"
    65   \<comment> \<open>This form displays the diagonal term.\<close>
    66   by best
    67 
    68 schematic_goal "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
    69   \<comment> \<open>This form exploits the set constructs.\<close>
    70   by (rule notI, erule rangeE, best)
    71 
    72 schematic_goal "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
    73   \<comment> \<open>Or just this!\<close>
    74   by best
    75 
    76 
    77 subsection \<open>The Schröder-Bernstein Theorem\<close>
    78 
    79 lemma disj_lemma: "- (f ` X) = g' ` (-X) \<Longrightarrow> f a = g' b \<Longrightarrow> a \<in> X \<Longrightarrow> b \<in> X"
    80   by blast
    81 
    82 lemma surj_if_then_else:
    83   "-(f ` X) = g' ` (-X) \<Longrightarrow> surj (\<lambda>z. if z \<in> X then f z else g' z)"
    84   by (simp add: surj_def) blast
    85 
    86 lemma bij_if_then_else:
    87   "inj_on f X \<Longrightarrow> inj_on g' (-X) \<Longrightarrow> -(f ` X) = g' ` (-X) \<Longrightarrow>
    88     h = (\<lambda>z. if z \<in> X then f z else g' z) \<Longrightarrow> inj h \<and> surj h"
    89   apply (unfold inj_on_def)
    90   apply (simp add: surj_if_then_else)
    91   apply (blast dest: disj_lemma sym)
    92   done
    93 
    94 lemma decomposition: "\<exists>X. X = - (g ` (- (f ` X)))"
    95   apply (rule exI)
    96   apply (rule lfp_unfold)
    97   apply (rule monoI, blast)
    98   done
    99 
   100 theorem Schroeder_Bernstein:
   101   "inj (f :: 'a \<Rightarrow> 'b) \<Longrightarrow> inj (g :: 'b \<Rightarrow> 'a)
   102     \<Longrightarrow> \<exists>h:: 'a \<Rightarrow> 'b. inj h \<and> surj h"
   103   apply (rule decomposition [where f=f and g=g, THEN exE])
   104   apply (rule_tac x = "(\<lambda>z. if z \<in> x then f z else inv g z)" in exI) 
   105     \<comment>\<open>The term above can be synthesized by a sufficiently detailed proof.\<close>
   106   apply (rule bij_if_then_else)
   107      apply (rule_tac [4] refl)
   108     apply (rule_tac [2] inj_on_inv_into)
   109     apply (erule subset_inj_on [OF _ subset_UNIV])
   110    apply blast
   111   apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric])
   112   done
   113 
   114 
   115 subsection \<open>A simple party theorem\<close>
   116 
   117 text\<open>\emph{At any party there are two people who know the same
   118 number of people}. Provided the party consists of at least two people
   119 and the knows relation is symmetric. Knowing yourself does not count
   120 --- otherwise knows needs to be reflexive. (From Freek Wiedijk's talk
   121 at TPHOLs 2007.)\<close>
   122 
   123 lemma equal_number_of_acquaintances:
   124 assumes "Domain R <= A" and "sym R" and "card A \<ge> 2"
   125 shows "\<not> inj_on (%a. card(R `` {a} - {a})) A"
   126 proof -
   127   let ?N = "%a. card(R `` {a} - {a})"
   128   let ?n = "card A"
   129   have "finite A" using \<open>card A \<ge> 2\<close> by(auto intro:ccontr)
   130   have 0: "R `` A <= A" using \<open>sym R\<close> \<open>Domain R <= A\<close>
   131     unfolding Domain_unfold sym_def by blast
   132   have h: "ALL a:A. R `` {a} <= A" using 0 by blast
   133   hence 1: "ALL a:A. finite(R `` {a})" using \<open>finite A\<close>
   134     by(blast intro: finite_subset)
   135   have sub: "?N ` A <= {0..<?n}"
   136   proof -
   137     have "ALL a:A. R `` {a} - {a} < A" using h by blast
   138     thus ?thesis using psubset_card_mono[OF \<open>finite A\<close>] by auto
   139   qed
   140   show "~ inj_on ?N A" (is "~ ?I")
   141   proof
   142     assume ?I
   143     hence "?n = card(?N ` A)" by(rule card_image[symmetric])
   144     with sub \<open>finite A\<close> have 2[simp]: "?N ` A = {0..<?n}"
   145       using subset_card_intvl_is_intvl[of _ 0] by(auto)
   146     have "0 : ?N ` A" and "?n - 1 : ?N ` A"  using \<open>card A \<ge> 2\<close> by simp+
   147     then obtain a b where ab: "a:A" "b:A" and Na: "?N a = 0" and Nb: "?N b = ?n - 1"
   148       by (auto simp del: 2)
   149     have "a \<noteq> b" using Na Nb \<open>card A \<ge> 2\<close> by auto
   150     have "R `` {a} - {a} = {}" by (metis 1 Na ab card_eq_0_iff finite_Diff)
   151     hence "b \<notin> R `` {a}" using \<open>a\<noteq>b\<close> by blast
   152     hence "a \<notin> R `` {b}" by (metis Image_singleton_iff assms(2) sym_def)
   153     hence 3: "R `` {b} - {b} <= A - {a,b}" using 0 ab by blast
   154     have 4: "finite (A - {a,b})" using \<open>finite A\<close> by simp
   155     have "?N b <= ?n - 2" using ab \<open>a\<noteq>b\<close> \<open>finite A\<close> card_mono[OF 4 3] by simp
   156     then show False using Nb \<open>card A \<ge>  2\<close> by arith
   157   qed
   158 qed
   159 
   160 text \<open>
   161   From W. W. Bledsoe and Guohui Feng, SET-VAR. JAR 11 (3), 1993, pages
   162   293-314.
   163 
   164   Isabelle can prove the easy examples without any special mechanisms,
   165   but it can't prove the hard ones.
   166 \<close>
   167 
   168 lemma "\<exists>A. (\<forall>x \<in> A. x \<le> (0::int))"
   169   \<comment> \<open>Example 1, page 295.\<close>
   170   by force
   171 
   172 lemma "D \<in> F \<Longrightarrow> \<exists>G. \<forall>A \<in> G. \<exists>B \<in> F. A \<subseteq> B"
   173   \<comment> \<open>Example 2.\<close>
   174   by force
   175 
   176 lemma "P a \<Longrightarrow> \<exists>A. (\<forall>x \<in> A. P x) \<and> (\<exists>y. y \<in> A)"
   177   \<comment> \<open>Example 3.\<close>
   178   by force
   179 
   180 lemma "a < b \<and> b < (c::int) \<Longrightarrow> \<exists>A. a \<notin> A \<and> b \<in> A \<and> c \<notin> A"
   181   \<comment> \<open>Example 4.\<close>
   182   by auto \<comment>\<open>slow\<close>
   183 
   184 lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
   185   \<comment> \<open>Example 5, page 298.\<close>
   186   by force
   187 
   188 lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
   189   \<comment> \<open>Example 6.\<close>
   190   by force
   191 
   192 lemma "\<exists>A. a \<notin> A"
   193   \<comment> \<open>Example 7.\<close>
   194   by force
   195 
   196 lemma "(\<forall>u v. u < (0::int) \<longrightarrow> u \<noteq> \<bar>v\<bar>)
   197     \<longrightarrow> (\<exists>A::int set. -2 \<in> A & (\<forall>y. \<bar>y\<bar> \<notin> A))"
   198   \<comment> \<open>Example 8 needs a small hint.\<close>
   199   by force
   200     \<comment> \<open>not \<open>blast\<close>, which can't simplify \<open>-2 < 0\<close>\<close>
   201 
   202 text \<open>Example 9 omitted (requires the reals).\<close>
   203 
   204 text \<open>The paper has no Example 10!\<close>
   205 
   206 lemma "(\<forall>A. 0 \<in> A \<and> (\<forall>x \<in> A. Suc x \<in> A) \<longrightarrow> n \<in> A) \<and>
   207   P 0 \<and> (\<forall>x. P x \<longrightarrow> P (Suc x)) \<longrightarrow> P n"
   208   \<comment> \<open>Example 11: needs a hint.\<close>
   209 by(metis nat.induct)
   210 
   211 lemma
   212   "(\<forall>A. (0, 0) \<in> A \<and> (\<forall>x y. (x, y) \<in> A \<longrightarrow> (Suc x, Suc y) \<in> A) \<longrightarrow> (n, m) \<in> A)
   213     \<and> P n \<longrightarrow> P m"
   214   \<comment> \<open>Example 12.\<close>
   215   by auto
   216 
   217 lemma
   218   "(\<forall>x. (\<exists>u. x = 2 * u) = (\<not> (\<exists>v. Suc x = 2 * v))) \<longrightarrow>
   219     (\<exists>A. \<forall>x. (x \<in> A) = (Suc x \<notin> A))"
   220   \<comment> \<open>Example EO1: typo in article, and with the obvious fix it seems
   221       to require arithmetic reasoning.\<close>
   222   apply clarify
   223   apply (rule_tac x = "{x. \<exists>u. x = 2 * u}" in exI, auto)
   224    apply metis+
   225   done
   226 
   227 end