src/HOL/ex/set.thy
 author paulson Fri Sep 14 15:27:12 2007 +0200 (2007-09-14) changeset 24573 5bbdc9b60648 parent 19982 e4d50f8f3722 child 24853 aab5798e5a33 permissions -rw-r--r--
tidied
```     1 (*  Title:      HOL/ex/set.thy
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```     2     ID:         \$Id\$
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```     3     Author:     Tobias Nipkow and Lawrence C Paulson
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```     4     Copyright   1991  University of Cambridge
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```     5 *)
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```     6
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```     7 header {* Set Theory examples: Cantor's Theorem, Schröder-Bernstein Theorem, etc. *}
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```     8
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```     9 theory set imports Main begin
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```    10
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```    11 text{*
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```    12   These two are cited in Benzmueller and Kohlhase's system description
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```    13   of LEO, CADE-15, 1998 (pages 139-143) as theorems LEO could not
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```    14   prove.
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```    15 *}
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```    16
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```    17 lemma "(X = Y \<union> Z) =
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```    18     (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"
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```    19   by blast
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```    20
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```    21 lemma "(X = Y \<inter> Z) =
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```    22     (X \<subseteq> Y \<and> X \<subseteq> Z \<and> (\<forall>V. V \<subseteq> Y \<and> V \<subseteq> Z \<longrightarrow> V \<subseteq> X))"
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```    23   by blast
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```    24
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```    25 text {*
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```    26   Trivial example of term synthesis: apparently hard for some provers!
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```    27 *}
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```    28
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```    29 lemma "a \<noteq> b \<Longrightarrow> a \<in> ?X \<and> b \<notin> ?X"
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```    30   by blast
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```    31
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```    32
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```    33 subsection {* Examples for the @{text blast} paper *}
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```    34
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```    35 lemma "(\<Union>x \<in> C. f x \<union> g x) = \<Union>(f ` C)  \<union>  \<Union>(g ` C)"
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```    36   -- {* Union-image, called @{text Un_Union_image} in Main HOL *}
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```    37   by blast
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```    38
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```    39 lemma "(\<Inter>x \<in> C. f x \<inter> g x) = \<Inter>(f ` C) \<inter> \<Inter>(g ` C)"
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```    40   -- {* Inter-image, called @{text Int_Inter_image} in Main HOL *}
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```    41   by blast
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```    42
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```    43 lemma singleton_example_1:
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```    44      "\<And>S::'a set set. \<forall>x \<in> S. \<forall>y \<in> S. x \<subseteq> y \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
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```    45   by blast
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```    46
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```    47 lemma singleton_example_2:
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```    48      "\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
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```    49   -- {*Variant of the problem above. *}
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```    50   by blast
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```    51
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```    52 lemma "\<exists>!x. f (g x) = x \<Longrightarrow> \<exists>!y. g (f y) = y"
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```    53   -- {* A unique fixpoint theorem --- @{text fast}/@{text best}/@{text meson} all fail. *}
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```    54   by metis
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```    55
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```    56
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```    57 subsection {* Cantor's Theorem: There is no surjection from a set to its powerset *}
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```    58
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```    59 lemma cantor1: "\<not> (\<exists>f:: 'a \<Rightarrow> 'a set. \<forall>S. \<exists>x. f x = S)"
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```    60   -- {* Requires best-first search because it is undirectional. *}
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```    61   by best
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```    62
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```    63 lemma "\<forall>f:: 'a \<Rightarrow> 'a set. \<forall>x. f x \<noteq> ?S f"
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```    64   -- {*This form displays the diagonal term. *}
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```    65   by best
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```    66
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```    67 lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
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```    68   -- {* This form exploits the set constructs. *}
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```    69   by (rule notI, erule rangeE, best)
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```    70
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```    71 lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
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```    72   -- {* Or just this! *}
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```    73   by best
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```    74
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```    75
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```    76 subsection {* The Schröder-Berstein Theorem *}
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```    77
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```    78 lemma disj_lemma: "- (f ` X) = g ` (-X) \<Longrightarrow> f a = g b \<Longrightarrow> a \<in> X \<Longrightarrow> b \<in> X"
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```    79   by blast
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```    80
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```    81 lemma surj_if_then_else:
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```    82   "-(f ` X) = g ` (-X) \<Longrightarrow> surj (\<lambda>z. if z \<in> X then f z else g z)"
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```    83   by (simp add: surj_def) blast
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```    84
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```    85 lemma bij_if_then_else:
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```    86   "inj_on f X \<Longrightarrow> inj_on g (-X) \<Longrightarrow> -(f ` X) = g ` (-X) \<Longrightarrow>
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```    87     h = (\<lambda>z. if z \<in> X then f z else g z) \<Longrightarrow> inj h \<and> surj h"
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```    88   apply (unfold inj_on_def)
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```    89   apply (simp add: surj_if_then_else)
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```    90   apply (blast dest: disj_lemma sym)
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```    91   done
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```    92
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```    93 lemma decomposition: "\<exists>X. X = - (g ` (- (f ` X)))"
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```    94   apply (rule exI)
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```    95   apply (rule lfp_unfold)
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```    96   apply (rule monoI, blast)
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```    97   done
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```    98
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```    99 theorem Schroeder_Bernstein:
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```   100   "inj (f :: 'a \<Rightarrow> 'b) \<Longrightarrow> inj (g :: 'b \<Rightarrow> 'a)
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```   101     \<Longrightarrow> \<exists>h:: 'a \<Rightarrow> 'b. inj h \<and> surj h"
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```   102   apply (rule decomposition [where f=f and g=g, THEN exE])
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```   103   apply (rule_tac x = "(\<lambda>z. if z \<in> x then f z else inv g z)" in exI)
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```   104     --{*The term above can be synthesized by a sufficiently detailed proof.*}
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```   105   apply (rule bij_if_then_else)
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```   106      apply (rule_tac [4] refl)
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```   107     apply (rule_tac [2] inj_on_inv)
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```   108     apply (erule subset_inj_on [OF _ subset_UNIV])
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```   109    apply blast
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```   110   apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric])
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```   111   done
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```   112
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```   113
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```   114 text {*
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```   115   From W. W. Bledsoe and Guohui Feng, SET-VAR. JAR 11 (3), 1993, pages
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```   116   293-314.
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```   117
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```   118   Isabelle can prove the easy examples without any special mechanisms,
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```   119   but it can't prove the hard ones.
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```   120 *}
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```   121
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```   122 lemma "\<exists>A. (\<forall>x \<in> A. x \<le> (0::int))"
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```   123   -- {* Example 1, page 295. *}
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```   124   by force
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```   125
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```   126 lemma "D \<in> F \<Longrightarrow> \<exists>G. \<forall>A \<in> G. \<exists>B \<in> F. A \<subseteq> B"
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```   127   -- {* Example 2. *}
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```   128   by force
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```   129
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```   130 lemma "P a \<Longrightarrow> \<exists>A. (\<forall>x \<in> A. P x) \<and> (\<exists>y. y \<in> A)"
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```   131   -- {* Example 3. *}
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```   132   by force
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```   133
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```   134 lemma "a < b \<and> b < (c::int) \<Longrightarrow> \<exists>A. a \<notin> A \<and> b \<in> A \<and> c \<notin> A"
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```   135   -- {* Example 4. *}
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```   136   by force
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```   137
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```   138 lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
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```   139   -- {*Example 5, page 298. *}
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```   140   by force
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```   141
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```   142 lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
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```   143   -- {* Example 6. *}
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```   144   by force
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```   145
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```   146 lemma "\<exists>A. a \<notin> A"
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```   147   -- {* Example 7. *}
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```   148   by force
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```   149
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```   150 lemma "(\<forall>u v. u < (0::int) \<longrightarrow> u \<noteq> abs v)
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```   151     \<longrightarrow> (\<exists>A::int set. (\<forall>y. abs y \<notin> A) \<and> -2 \<in> A)"
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```   152   -- {* Example 8 now needs a small hint. *}
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```   153   by (simp add: abs_if, force)
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```   154     -- {* not @{text blast}, which can't simplify @{text "-2 < 0"} *}
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```   155
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```   156 text {* Example 9 omitted (requires the reals). *}
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```   157
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```   158 text {* The paper has no Example 10! *}
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```   159
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```   160 lemma "(\<forall>A. 0 \<in> A \<and> (\<forall>x \<in> A. Suc x \<in> A) \<longrightarrow> n \<in> A) \<and>
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```   161   P 0 \<and> (\<forall>x. P x \<longrightarrow> P (Suc x)) \<longrightarrow> P n"
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```   162   -- {* Example 11: needs a hint. *}
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```   163   apply clarify
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```   164   apply (drule_tac x = "{x. P x}" in spec)
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```   165   apply force
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```   166   done
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```   167
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```   168 lemma
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```   169   "(\<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)
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```   170     \<and> P n \<longrightarrow> P m"
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```   171   -- {* Example 12. *}
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```   172   by auto
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```   173
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```   174 lemma
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```   175   "(\<forall>x. (\<exists>u. x = 2 * u) = (\<not> (\<exists>v. Suc x = 2 * v))) \<longrightarrow>
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```   176     (\<exists>A. \<forall>x. (x \<in> A) = (Suc x \<notin> A))"
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```   177   -- {* Example EO1: typo in article, and with the obvious fix it seems
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```   178       to require arithmetic reasoning. *}
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```   179   apply clarify
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```   180   apply (rule_tac x = "{x. \<exists>u. x = 2 * u}" in exI, auto)
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```   181    apply (case_tac v, auto)
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```   182   apply (drule_tac x = "Suc v" and P = "\<lambda>x. ?a x \<noteq> ?b x" in spec, force)
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```   183   done
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```   184
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```   185 end
```