src/HOL/ex/set.thy
 author nipkow Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) changeset 30313 b2441b0c8d38 parent 24853 aab5798e5a33 child 32988 d1d4d7a08a66 permissions -rw-r--r--
     1 (*  Title:      HOL/ex/set.thy

     2     ID:         $Id$

     3     Author:     Tobias Nipkow and Lawrence C Paulson

     4     Copyright   1991  University of Cambridge

     5 *)

     6

     7 header {* Set Theory examples: Cantor's Theorem, Schröder-Bernstein Theorem, etc. *}

     8

     9 theory set imports Main begin

    10

    11 text{*

    12   These two are cited in Benzmueller and Kohlhase's system description

    13   of LEO, CADE-15, 1998 (pages 139-143) as theorems LEO could not

    14   prove.

    15 *}

    16

    17 lemma "(X = Y \<union> Z) =

    18     (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"

    19   by blast

    20

    21 lemma "(X = Y \<inter> Z) =

    22     (X \<subseteq> Y \<and> X \<subseteq> Z \<and> (\<forall>V. V \<subseteq> Y \<and> V \<subseteq> Z \<longrightarrow> V \<subseteq> X))"

    23   by blast

    24

    25 text {*

    26   Trivial example of term synthesis: apparently hard for some provers!

    27 *}

    28

    29 lemma "a \<noteq> b \<Longrightarrow> a \<in> ?X \<and> b \<notin> ?X"

    30   by blast

    31

    32

    33 subsection {* Examples for the @{text blast} paper *}

    34

    35 lemma "(\<Union>x \<in> C. f x \<union> g x) = \<Union>(f  C)  \<union>  \<Union>(g  C)"

    36   -- {* Union-image, called @{text Un_Union_image} in Main HOL *}

    37   by blast

    38

    39 lemma "(\<Inter>x \<in> C. f x \<inter> g x) = \<Inter>(f  C) \<inter> \<Inter>(g  C)"

    40   -- {* Inter-image, called @{text Int_Inter_image} in Main HOL *}

    41   by blast

    42

    43 lemma singleton_example_1:

    44      "\<And>S::'a set set. \<forall>x \<in> S. \<forall>y \<in> S. x \<subseteq> y \<Longrightarrow> \<exists>z. S \<subseteq> {z}"

    45   by blast

    46

    47 lemma singleton_example_2:

    48      "\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}"

    49   -- {*Variant of the problem above. *}

    50   by blast

    51

    52 lemma "\<exists>!x. f (g x) = x \<Longrightarrow> \<exists>!y. g (f y) = y"

    53   -- {* A unique fixpoint theorem --- @{text fast}/@{text best}/@{text meson} all fail. *}

    54   by metis

    55

    56

    57 subsection {* Cantor's Theorem: There is no surjection from a set to its powerset *}

    58

    59 lemma cantor1: "\<not> (\<exists>f:: 'a \<Rightarrow> 'a set. \<forall>S. \<exists>x. f x = S)"

    60   -- {* Requires best-first search because it is undirectional. *}

    61   by best

    62

    63 lemma "\<forall>f:: 'a \<Rightarrow> 'a set. \<forall>x. f x \<noteq> ?S f"

    64   -- {*This form displays the diagonal term. *}

    65   by best

    66

    67 lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"

    68   -- {* This form exploits the set constructs. *}

    69   by (rule notI, erule rangeE, best)

    70

    71 lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"

    72   -- {* Or just this! *}

    73   by best

    74

    75

    76 subsection {* The Schröder-Berstein Theorem *}

    77

    78 lemma disj_lemma: "- (f  X) = g  (-X) \<Longrightarrow> f a = g b \<Longrightarrow> a \<in> X \<Longrightarrow> b \<in> X"

    79   by blast

    80

    81 lemma surj_if_then_else:

    82   "-(f  X) = g  (-X) \<Longrightarrow> surj (\<lambda>z. if z \<in> X then f z else g z)"

    83   by (simp add: surj_def) blast

    84

    85 lemma bij_if_then_else:

    86   "inj_on f X \<Longrightarrow> inj_on g (-X) \<Longrightarrow> -(f  X) = g  (-X) \<Longrightarrow>

    87     h = (\<lambda>z. if z \<in> X then f z else g z) \<Longrightarrow> inj h \<and> surj h"

    88   apply (unfold inj_on_def)

    89   apply (simp add: surj_if_then_else)

    90   apply (blast dest: disj_lemma sym)

    91   done

    92

    93 lemma decomposition: "\<exists>X. X = - (g  (- (f  X)))"

    94   apply (rule exI)

    95   apply (rule lfp_unfold)

    96   apply (rule monoI, blast)

    97   done

    98

    99 theorem Schroeder_Bernstein:

   100   "inj (f :: 'a \<Rightarrow> 'b) \<Longrightarrow> inj (g :: 'b \<Rightarrow> 'a)

   101     \<Longrightarrow> \<exists>h:: 'a \<Rightarrow> 'b. inj h \<and> surj h"

   102   apply (rule decomposition [where f=f and g=g, THEN exE])

   103   apply (rule_tac x = "(\<lambda>z. if z \<in> x then f z else inv g z)" in exI)

   104     --{*The term above can be synthesized by a sufficiently detailed proof.*}

   105   apply (rule bij_if_then_else)

   106      apply (rule_tac  refl)

   107     apply (rule_tac  inj_on_inv)

   108     apply (erule subset_inj_on [OF _ subset_UNIV])

   109    apply blast

   110   apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric])

   111   done

   112

   113

   114 subsection {* A simple party theorem *}

   115

   116 text{* \emph{At any party there are two people who know the same

   117 number of people}. Provided the party consists of at least two people

   118 and the knows relation is symmetric. Knowing yourself does not count

   119 --- otherwise knows needs to be reflexive. (From Freek Wiedijk's talk

   120 at TPHOLs 2007.) *}

   121

   122 lemma equal_number_of_acquaintances:

   123 assumes "Domain R <= A" and "sym R" and "card A \<ge> 2"

   124 shows "\<not> inj_on (%a. card(R  {a} - {a})) A"

   125 proof -

   126   let ?N = "%a. card(R  {a} - {a})"

   127   let ?n = "card A"

   128   have "finite A" using card A \<ge> 2 by(auto intro:ccontr)

   129   have 0: "R  A <= A" using sym R Domain R <= A

   130     unfolding Domain_def sym_def by blast

   131   have h: "ALL a:A. R  {a} <= A" using 0 by blast

   132   hence 1: "ALL a:A. finite(R  {a})" using finite A

   133     by(blast intro: finite_subset)

   134   have sub: "?N  A <= {0..<?n}"

   135   proof -

   136     have "ALL a:A. R  {a} - {a} < A" using h by blast

   137     thus ?thesis using psubset_card_mono[OF finite A] by auto

   138   qed

   139   show "~ inj_on ?N A" (is "~ ?I")

   140   proof

   141     assume ?I

   142     hence "?n = card(?N  A)" by(rule card_image[symmetric])

   143     with sub finite A have 2[simp]: "?N  A = {0..<?n}"

   144       using subset_card_intvl_is_intvl[of _ 0] by(auto)

   145     have "0 : ?N  A" and "?n - 1 : ?N  A"  using card A \<ge> 2 by simp+

   146     then obtain a b where ab: "a:A" "b:A" and Na: "?N a = 0" and Nb: "?N b = ?n - 1"

   147       by (auto simp del: 2)

   148     have "a \<noteq> b" using Na Nb card A \<ge> 2 by auto

   149     have "R  {a} - {a} = {}" by (metis 1 Na ab card_eq_0_iff finite_Diff)

   150     hence "b \<notin> R  {a}" using a\<noteq>b by blast

   151     hence "a \<notin> R  {b}" by (metis Image_singleton_iff assms(2) sym_def)

   152     hence 3: "R  {b} - {b} <= A - {a,b}" using 0 ab by blast

   153     have 4: "finite (A - {a,b})" using finite A by simp

   154     have "?N b <= ?n - 2" using ab a\<noteq>b finite A card_mono[OF 4 3] by simp

   155     then show False using Nb card A \<ge>  2 by arith

   156   qed

   157 qed

   158

   159 text {*

   160   From W. W. Bledsoe and Guohui Feng, SET-VAR. JAR 11 (3), 1993, pages

   161   293-314.

   162

   163   Isabelle can prove the easy examples without any special mechanisms,

   164   but it can't prove the hard ones.

   165 *}

   166

   167 lemma "\<exists>A. (\<forall>x \<in> A. x \<le> (0::int))"

   168   -- {* Example 1, page 295. *}

   169   by force

   170

   171 lemma "D \<in> F \<Longrightarrow> \<exists>G. \<forall>A \<in> G. \<exists>B \<in> F. A \<subseteq> B"

   172   -- {* Example 2. *}

   173   by force

   174

   175 lemma "P a \<Longrightarrow> \<exists>A. (\<forall>x \<in> A. P x) \<and> (\<exists>y. y \<in> A)"

   176   -- {* Example 3. *}

   177   by force

   178

   179 lemma "a < b \<and> b < (c::int) \<Longrightarrow> \<exists>A. a \<notin> A \<and> b \<in> A \<and> c \<notin> A"

   180   -- {* Example 4. *}

   181   by force

   182

   183 lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"

   184   -- {*Example 5, page 298. *}

   185   by force

   186

   187 lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"

   188   -- {* Example 6. *}

   189   by force

   190

   191 lemma "\<exists>A. a \<notin> A"

   192   -- {* Example 7. *}

   193   by force

   194

   195 lemma "(\<forall>u v. u < (0::int) \<longrightarrow> u \<noteq> abs v)

   196     \<longrightarrow> (\<exists>A::int set. (\<forall>y. abs y \<notin> A) \<and> -2 \<in> A)"

   197   -- {* Example 8 now needs a small hint. *}

   198   by (simp add: abs_if, force)

   199     -- {* not @{text blast}, which can't simplify @{text "-2 < 0"} *}

   200

   201 text {* Example 9 omitted (requires the reals). *}

   202

   203 text {* The paper has no Example 10! *}

   204

   205 lemma "(\<forall>A. 0 \<in> A \<and> (\<forall>x \<in> A. Suc x \<in> A) \<longrightarrow> n \<in> A) \<and>

   206   P 0 \<and> (\<forall>x. P x \<longrightarrow> P (Suc x)) \<longrightarrow> P n"

   207   -- {* Example 11: needs a hint. *}

   208   apply clarify

   209   apply (drule_tac x = "{x. P x}" in spec)

   210   apply force

   211   done

   212

   213 lemma

   214   "(\<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)

   215     \<and> P n \<longrightarrow> P m"

   216   -- {* Example 12. *}

   217   by auto

   218

   219 lemma

   220   "(\<forall>x. (\<exists>u. x = 2 * u) = (\<not> (\<exists>v. Suc x = 2 * v))) \<longrightarrow>

   221     (\<exists>A. \<forall>x. (x \<in> A) = (Suc x \<notin> A))"

   222   -- {* Example EO1: typo in article, and with the obvious fix it seems

   223       to require arithmetic reasoning. *}

   224   apply clarify

   225   apply (rule_tac x = "{x. \<exists>u. x = 2 * u}" in exI, auto)

   226    apply (case_tac v, auto)

   227   apply (drule_tac x = "Suc v" and P = "\<lambda>x. ?a x \<noteq> ?b x" in spec, force)

   228   done

   229

   230 end
`