avoid case-sensitive name for example theory
authorhaftmann
Thu Aug 18 13:10:24 2011 +0200 (2011-08-18)
changeset 44276fe769a0fcc96
parent 44263 971d1be5d5ce
child 44277 bcb696533579
avoid case-sensitive name for example theory
src/HOL/IsaMakefile
src/HOL/ex/ROOT.ML
src/HOL/ex/Set_Theory.thy
src/HOL/ex/set.thy
     1.1 --- a/src/HOL/IsaMakefile	Thu Aug 18 16:52:19 2011 +0900
     1.2 +++ b/src/HOL/IsaMakefile	Thu Aug 18 13:10:24 2011 +0200
     1.3 @@ -1039,32 +1039,30 @@
     1.4  
     1.5  $(LOG)/HOL-ex.gz: $(OUT)/HOL Decision_Procs/Commutative_Ring.thy	\
     1.6    Number_Theory/Primes.thy ex/Abstract_NAT.thy ex/Antiquote.thy		\
     1.7 -  ex/Arith_Examples.thy ex/Arithmetic_Series_Complex.thy \
     1.8 -  ex/BT.thy	ex/BinEx.thy ex/Binary.thy ex/Birthday_Paradox.thy			\
     1.9 -  ex/CTL.thy ex/Case_Product.thy			\
    1.10 -  ex/Chinese.thy ex/Classical.thy ex/CodegenSML_Test.thy		\
    1.11 -  ex/Coercion_Examples.thy ex/Coherent.thy ex/Dedekind_Real.thy		\
    1.12 -  ex/Efficient_Nat_examples.thy ex/Eval_Examples.thy ex/Fundefs.thy	\
    1.13 -  ex/Gauge_Integration.thy ex/Groebner_Examples.thy ex/Guess.thy	\
    1.14 -  ex/HarmonicSeries.thy ex/Hebrew.thy ex/Hex_Bin_Examples.thy		\
    1.15 -  ex/Higher_Order_Logic.thy ex/Iff_Oracle.thy ex/Induction_Schema.thy	\
    1.16 +  ex/Arith_Examples.thy ex/Arithmetic_Series_Complex.thy ex/BT.thy	\
    1.17 +  ex/BinEx.thy ex/Binary.thy ex/Birthday_Paradox.thy ex/CTL.thy		\
    1.18 +  ex/Case_Product.thy ex/Chinese.thy ex/Classical.thy			\
    1.19 +  ex/CodegenSML_Test.thy ex/Coercion_Examples.thy ex/Coherent.thy	\
    1.20 +  ex/Dedekind_Real.thy ex/Efficient_Nat_examples.thy			\
    1.21 +  ex/Eval_Examples.thy ex/Fundefs.thy ex/Gauge_Integration.thy		\
    1.22 +  ex/Groebner_Examples.thy ex/Guess.thy ex/HarmonicSeries.thy		\
    1.23 +  ex/Hebrew.thy ex/Hex_Bin_Examples.thy ex/Higher_Order_Logic.thy	\
    1.24 +  ex/Iff_Oracle.thy ex/Induction_Schema.thy				\
    1.25    ex/Interpretation_with_Defs.thy ex/Intuitionistic.thy ex/Lagrange.thy	\
    1.26    ex/List_to_Set_Comprehension_Examples.thy ex/LocaleTest2.thy		\
    1.27    ex/MT.thy ex/MergeSort.thy ex/Meson_Test.thy ex/MonoidGroup.thy	\
    1.28    ex/Multiquote.thy ex/NatSum.thy ex/Normalization_by_Evaluation.thy	\
    1.29    ex/Numeral.thy ex/PER.thy ex/PresburgerEx.thy ex/Primrec.thy		\
    1.30    ex/Quickcheck_Examples.thy ex/Quickcheck_Lattice_Examples.thy		\
    1.31 -  ex/Quickcheck_Narrowing_Examples.thy  				\
    1.32 -  ex/Quicksort.thy ex/ROOT.ML ex/Records.thy		\
    1.33 -  ex/ReflectionEx.thy ex/Refute_Examples.thy ex/SAT_Examples.thy	\
    1.34 -  ex/SVC_Oracle.thy ex/Serbian.thy ex/Set_Algebras.thy			\
    1.35 -  ex/sledgehammer_tactics.ML ex/Sqrt.thy				\
    1.36 -  ex/Sqrt_Script.thy ex/Sudoku.thy ex/Tarski.thy ex/Termination.thy	\
    1.37 -  ex/Transfer_Ex.thy ex/Tree23.thy			\
    1.38 +  ex/Quickcheck_Narrowing_Examples.thy ex/Quicksort.thy ex/ROOT.ML	\
    1.39 +  ex/Records.thy ex/ReflectionEx.thy ex/Refute_Examples.thy		\
    1.40 +  ex/SAT_Examples.thy ex/Serbian.thy ex/Set_Theory.thy			\
    1.41 +  ex/Set_Algebras.thy ex/SVC_Oracle.thy ex/sledgehammer_tactics.ML	\
    1.42 +  ex/Sqrt.thy ex/Sqrt_Script.thy ex/Sudoku.thy ex/Tarski.thy		\
    1.43 +  ex/Termination.thy ex/Transfer_Ex.thy ex/Tree23.thy			\
    1.44    ex/Unification.thy ex/While_Combinator_Example.thy			\
    1.45 -  ex/document/root.bib ex/document/root.tex ex/set.thy ex/svc_funcs.ML	\
    1.46 -  ex/svc_test.thy							\
    1.47 -  ../Tools/interpretation_with_defs.ML
    1.48 +  ex/document/root.bib ex/document/root.tex ex/svc_funcs.ML		\
    1.49 +  ex/svc_test.thy ../Tools/interpretation_with_defs.ML
    1.50  	@$(ISABELLE_TOOL) usedir $(OUT)/HOL ex
    1.51  
    1.52  
     2.1 --- a/src/HOL/ex/ROOT.ML	Thu Aug 18 16:52:19 2011 +0900
     2.2 +++ b/src/HOL/ex/ROOT.ML	Thu Aug 18 13:10:24 2011 +0200
     2.3 @@ -48,7 +48,7 @@
     2.4    "Primrec",
     2.5    "Tarski",
     2.6    "Classical",
     2.7 -  "set",
     2.8 +  "Set_Theory",
     2.9    "Meson_Test",
    2.10    "Termination",
    2.11    "Coherent",
     3.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     3.2 +++ b/src/HOL/ex/Set_Theory.thy	Thu Aug 18 13:10:24 2011 +0200
     3.3 @@ -0,0 +1,227 @@
     3.4 +(*  Title:      HOL/ex/Set_Theory.thy
     3.5 +    Author:     Tobias Nipkow and Lawrence C Paulson
     3.6 +    Copyright   1991  University of Cambridge
     3.7 +*)
     3.8 +
     3.9 +header {* Set Theory examples: Cantor's Theorem, Schröder-Bernstein Theorem, etc. *}
    3.10 +
    3.11 +theory Set_Theory
    3.12 +imports Main
    3.13 +begin
    3.14 +
    3.15 +text{*
    3.16 +  These two are cited in Benzmueller and Kohlhase's system description
    3.17 +  of LEO, CADE-15, 1998 (pages 139-143) as theorems LEO could not
    3.18 +  prove.
    3.19 +*}
    3.20 +
    3.21 +lemma "(X = Y \<union> Z) =
    3.22 +    (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"
    3.23 +  by blast
    3.24 +
    3.25 +lemma "(X = Y \<inter> Z) =
    3.26 +    (X \<subseteq> Y \<and> X \<subseteq> Z \<and> (\<forall>V. V \<subseteq> Y \<and> V \<subseteq> Z \<longrightarrow> V \<subseteq> X))"
    3.27 +  by blast
    3.28 +
    3.29 +text {*
    3.30 +  Trivial example of term synthesis: apparently hard for some provers!
    3.31 +*}
    3.32 +
    3.33 +schematic_lemma "a \<noteq> b \<Longrightarrow> a \<in> ?X \<and> b \<notin> ?X"
    3.34 +  by blast
    3.35 +
    3.36 +
    3.37 +subsection {* Examples for the @{text blast} paper *}
    3.38 +
    3.39 +lemma "(\<Union>x \<in> C. f x \<union> g x) = \<Union>(f ` C)  \<union>  \<Union>(g ` C)"
    3.40 +  -- {* Union-image, called @{text Un_Union_image} in Main HOL *}
    3.41 +  by blast
    3.42 +
    3.43 +lemma "(\<Inter>x \<in> C. f x \<inter> g x) = \<Inter>(f ` C) \<inter> \<Inter>(g ` C)"
    3.44 +  -- {* Inter-image, called @{text Int_Inter_image} in Main HOL *}
    3.45 +  by blast
    3.46 +
    3.47 +lemma singleton_example_1:
    3.48 +     "\<And>S::'a set set. \<forall>x \<in> S. \<forall>y \<in> S. x \<subseteq> y \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
    3.49 +  by blast
    3.50 +
    3.51 +lemma singleton_example_2:
    3.52 +     "\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
    3.53 +  -- {*Variant of the problem above. *}
    3.54 +  by blast
    3.55 +
    3.56 +lemma "\<exists>!x. f (g x) = x \<Longrightarrow> \<exists>!y. g (f y) = y"
    3.57 +  -- {* A unique fixpoint theorem --- @{text fast}/@{text best}/@{text meson} all fail. *}
    3.58 +  by metis
    3.59 +
    3.60 +
    3.61 +subsection {* Cantor's Theorem: There is no surjection from a set to its powerset *}
    3.62 +
    3.63 +lemma cantor1: "\<not> (\<exists>f:: 'a \<Rightarrow> 'a set. \<forall>S. \<exists>x. f x = S)"
    3.64 +  -- {* Requires best-first search because it is undirectional. *}
    3.65 +  by best
    3.66 +
    3.67 +schematic_lemma "\<forall>f:: 'a \<Rightarrow> 'a set. \<forall>x. f x \<noteq> ?S f"
    3.68 +  -- {*This form displays the diagonal term. *}
    3.69 +  by best
    3.70 +
    3.71 +schematic_lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
    3.72 +  -- {* This form exploits the set constructs. *}
    3.73 +  by (rule notI, erule rangeE, best)
    3.74 +
    3.75 +schematic_lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
    3.76 +  -- {* Or just this! *}
    3.77 +  by best
    3.78 +
    3.79 +
    3.80 +subsection {* The Schröder-Berstein Theorem *}
    3.81 +
    3.82 +lemma disj_lemma: "- (f ` X) = g ` (-X) \<Longrightarrow> f a = g b \<Longrightarrow> a \<in> X \<Longrightarrow> b \<in> X"
    3.83 +  by blast
    3.84 +
    3.85 +lemma surj_if_then_else:
    3.86 +  "-(f ` X) = g ` (-X) \<Longrightarrow> surj (\<lambda>z. if z \<in> X then f z else g z)"
    3.87 +  by (simp add: surj_def) blast
    3.88 +
    3.89 +lemma bij_if_then_else:
    3.90 +  "inj_on f X \<Longrightarrow> inj_on g (-X) \<Longrightarrow> -(f ` X) = g ` (-X) \<Longrightarrow>
    3.91 +    h = (\<lambda>z. if z \<in> X then f z else g z) \<Longrightarrow> inj h \<and> surj h"
    3.92 +  apply (unfold inj_on_def)
    3.93 +  apply (simp add: surj_if_then_else)
    3.94 +  apply (blast dest: disj_lemma sym)
    3.95 +  done
    3.96 +
    3.97 +lemma decomposition: "\<exists>X. X = - (g ` (- (f ` X)))"
    3.98 +  apply (rule exI)
    3.99 +  apply (rule lfp_unfold)
   3.100 +  apply (rule monoI, blast)
   3.101 +  done
   3.102 +
   3.103 +theorem Schroeder_Bernstein:
   3.104 +  "inj (f :: 'a \<Rightarrow> 'b) \<Longrightarrow> inj (g :: 'b \<Rightarrow> 'a)
   3.105 +    \<Longrightarrow> \<exists>h:: 'a \<Rightarrow> 'b. inj h \<and> surj h"
   3.106 +  apply (rule decomposition [where f=f and g=g, THEN exE])
   3.107 +  apply (rule_tac x = "(\<lambda>z. if z \<in> x then f z else inv g z)" in exI) 
   3.108 +    --{*The term above can be synthesized by a sufficiently detailed proof.*}
   3.109 +  apply (rule bij_if_then_else)
   3.110 +     apply (rule_tac [4] refl)
   3.111 +    apply (rule_tac [2] inj_on_inv_into)
   3.112 +    apply (erule subset_inj_on [OF _ subset_UNIV])
   3.113 +   apply blast
   3.114 +  apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric])
   3.115 +  done
   3.116 +
   3.117 +
   3.118 +subsection {* A simple party theorem *}
   3.119 +
   3.120 +text{* \emph{At any party there are two people who know the same
   3.121 +number of people}. Provided the party consists of at least two people
   3.122 +and the knows relation is symmetric. Knowing yourself does not count
   3.123 +--- otherwise knows needs to be reflexive. (From Freek Wiedijk's talk
   3.124 +at TPHOLs 2007.) *}
   3.125 +
   3.126 +lemma equal_number_of_acquaintances:
   3.127 +assumes "Domain R <= A" and "sym R" and "card A \<ge> 2"
   3.128 +shows "\<not> inj_on (%a. card(R `` {a} - {a})) A"
   3.129 +proof -
   3.130 +  let ?N = "%a. card(R `` {a} - {a})"
   3.131 +  let ?n = "card A"
   3.132 +  have "finite A" using `card A \<ge> 2` by(auto intro:ccontr)
   3.133 +  have 0: "R `` A <= A" using `sym R` `Domain R <= A`
   3.134 +    unfolding Domain_def sym_def by blast
   3.135 +  have h: "ALL a:A. R `` {a} <= A" using 0 by blast
   3.136 +  hence 1: "ALL a:A. finite(R `` {a})" using `finite A`
   3.137 +    by(blast intro: finite_subset)
   3.138 +  have sub: "?N ` A <= {0..<?n}"
   3.139 +  proof -
   3.140 +    have "ALL a:A. R `` {a} - {a} < A" using h by blast
   3.141 +    thus ?thesis using psubset_card_mono[OF `finite A`] by auto
   3.142 +  qed
   3.143 +  show "~ inj_on ?N A" (is "~ ?I")
   3.144 +  proof
   3.145 +    assume ?I
   3.146 +    hence "?n = card(?N ` A)" by(rule card_image[symmetric])
   3.147 +    with sub `finite A` have 2[simp]: "?N ` A = {0..<?n}"
   3.148 +      using subset_card_intvl_is_intvl[of _ 0] by(auto)
   3.149 +    have "0 : ?N ` A" and "?n - 1 : ?N ` A"  using `card A \<ge> 2` by simp+
   3.150 +    then obtain a b where ab: "a:A" "b:A" and Na: "?N a = 0" and Nb: "?N b = ?n - 1"
   3.151 +      by (auto simp del: 2)
   3.152 +    have "a \<noteq> b" using Na Nb `card A \<ge> 2` by auto
   3.153 +    have "R `` {a} - {a} = {}" by (metis 1 Na ab card_eq_0_iff finite_Diff)
   3.154 +    hence "b \<notin> R `` {a}" using `a\<noteq>b` by blast
   3.155 +    hence "a \<notin> R `` {b}" by (metis Image_singleton_iff assms(2) sym_def)
   3.156 +    hence 3: "R `` {b} - {b} <= A - {a,b}" using 0 ab by blast
   3.157 +    have 4: "finite (A - {a,b})" using `finite A` by simp
   3.158 +    have "?N b <= ?n - 2" using ab `a\<noteq>b` `finite A` card_mono[OF 4 3] by simp
   3.159 +    then show False using Nb `card A \<ge>  2` by arith
   3.160 +  qed
   3.161 +qed
   3.162 +
   3.163 +text {*
   3.164 +  From W. W. Bledsoe and Guohui Feng, SET-VAR. JAR 11 (3), 1993, pages
   3.165 +  293-314.
   3.166 +
   3.167 +  Isabelle can prove the easy examples without any special mechanisms,
   3.168 +  but it can't prove the hard ones.
   3.169 +*}
   3.170 +
   3.171 +lemma "\<exists>A. (\<forall>x \<in> A. x \<le> (0::int))"
   3.172 +  -- {* Example 1, page 295. *}
   3.173 +  by force
   3.174 +
   3.175 +lemma "D \<in> F \<Longrightarrow> \<exists>G. \<forall>A \<in> G. \<exists>B \<in> F. A \<subseteq> B"
   3.176 +  -- {* Example 2. *}
   3.177 +  by force
   3.178 +
   3.179 +lemma "P a \<Longrightarrow> \<exists>A. (\<forall>x \<in> A. P x) \<and> (\<exists>y. y \<in> A)"
   3.180 +  -- {* Example 3. *}
   3.181 +  by force
   3.182 +
   3.183 +lemma "a < b \<and> b < (c::int) \<Longrightarrow> \<exists>A. a \<notin> A \<and> b \<in> A \<and> c \<notin> A"
   3.184 +  -- {* Example 4. *}
   3.185 +  by force
   3.186 +
   3.187 +lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
   3.188 +  -- {*Example 5, page 298. *}
   3.189 +  by force
   3.190 +
   3.191 +lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
   3.192 +  -- {* Example 6. *}
   3.193 +  by force
   3.194 +
   3.195 +lemma "\<exists>A. a \<notin> A"
   3.196 +  -- {* Example 7. *}
   3.197 +  by force
   3.198 +
   3.199 +lemma "(\<forall>u v. u < (0::int) \<longrightarrow> u \<noteq> abs v)
   3.200 +    \<longrightarrow> (\<exists>A::int set. (\<forall>y. abs y \<notin> A) \<and> -2 \<in> A)"
   3.201 +  -- {* Example 8 now needs a small hint. *}
   3.202 +  by (simp add: abs_if, force)
   3.203 +    -- {* not @{text blast}, which can't simplify @{text "-2 < 0"} *}
   3.204 +
   3.205 +text {* Example 9 omitted (requires the reals). *}
   3.206 +
   3.207 +text {* The paper has no Example 10! *}
   3.208 +
   3.209 +lemma "(\<forall>A. 0 \<in> A \<and> (\<forall>x \<in> A. Suc x \<in> A) \<longrightarrow> n \<in> A) \<and>
   3.210 +  P 0 \<and> (\<forall>x. P x \<longrightarrow> P (Suc x)) \<longrightarrow> P n"
   3.211 +  -- {* Example 11: needs a hint. *}
   3.212 +by(metis nat.induct)
   3.213 +
   3.214 +lemma
   3.215 +  "(\<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)
   3.216 +    \<and> P n \<longrightarrow> P m"
   3.217 +  -- {* Example 12. *}
   3.218 +  by auto
   3.219 +
   3.220 +lemma
   3.221 +  "(\<forall>x. (\<exists>u. x = 2 * u) = (\<not> (\<exists>v. Suc x = 2 * v))) \<longrightarrow>
   3.222 +    (\<exists>A. \<forall>x. (x \<in> A) = (Suc x \<notin> A))"
   3.223 +  -- {* Example EO1: typo in article, and with the obvious fix it seems
   3.224 +      to require arithmetic reasoning. *}
   3.225 +  apply clarify
   3.226 +  apply (rule_tac x = "{x. \<exists>u. x = 2 * u}" in exI, auto)
   3.227 +   apply metis+
   3.228 +  done
   3.229 +
   3.230 +end
     4.1 --- a/src/HOL/ex/set.thy	Thu Aug 18 16:52:19 2011 +0900
     4.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     4.3 @@ -1,225 +0,0 @@
     4.4 -(*  Title:      HOL/ex/set.thy
     4.5 -    Author:     Tobias Nipkow and Lawrence C Paulson
     4.6 -    Copyright   1991  University of Cambridge
     4.7 -*)
     4.8 -
     4.9 -header {* Set Theory examples: Cantor's Theorem, Schröder-Bernstein Theorem, etc. *}
    4.10 -
    4.11 -theory set imports Main begin
    4.12 -
    4.13 -text{*
    4.14 -  These two are cited in Benzmueller and Kohlhase's system description
    4.15 -  of LEO, CADE-15, 1998 (pages 139-143) as theorems LEO could not
    4.16 -  prove.
    4.17 -*}
    4.18 -
    4.19 -lemma "(X = Y \<union> Z) =
    4.20 -    (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"
    4.21 -  by blast
    4.22 -
    4.23 -lemma "(X = Y \<inter> Z) =
    4.24 -    (X \<subseteq> Y \<and> X \<subseteq> Z \<and> (\<forall>V. V \<subseteq> Y \<and> V \<subseteq> Z \<longrightarrow> V \<subseteq> X))"
    4.25 -  by blast
    4.26 -
    4.27 -text {*
    4.28 -  Trivial example of term synthesis: apparently hard for some provers!
    4.29 -*}
    4.30 -
    4.31 -schematic_lemma "a \<noteq> b \<Longrightarrow> a \<in> ?X \<and> b \<notin> ?X"
    4.32 -  by blast
    4.33 -
    4.34 -
    4.35 -subsection {* Examples for the @{text blast} paper *}
    4.36 -
    4.37 -lemma "(\<Union>x \<in> C. f x \<union> g x) = \<Union>(f ` C)  \<union>  \<Union>(g ` C)"
    4.38 -  -- {* Union-image, called @{text Un_Union_image} in Main HOL *}
    4.39 -  by blast
    4.40 -
    4.41 -lemma "(\<Inter>x \<in> C. f x \<inter> g x) = \<Inter>(f ` C) \<inter> \<Inter>(g ` C)"
    4.42 -  -- {* Inter-image, called @{text Int_Inter_image} in Main HOL *}
    4.43 -  by blast
    4.44 -
    4.45 -lemma singleton_example_1:
    4.46 -     "\<And>S::'a set set. \<forall>x \<in> S. \<forall>y \<in> S. x \<subseteq> y \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
    4.47 -  by blast
    4.48 -
    4.49 -lemma singleton_example_2:
    4.50 -     "\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
    4.51 -  -- {*Variant of the problem above. *}
    4.52 -  by blast
    4.53 -
    4.54 -lemma "\<exists>!x. f (g x) = x \<Longrightarrow> \<exists>!y. g (f y) = y"
    4.55 -  -- {* A unique fixpoint theorem --- @{text fast}/@{text best}/@{text meson} all fail. *}
    4.56 -  by metis
    4.57 -
    4.58 -
    4.59 -subsection {* Cantor's Theorem: There is no surjection from a set to its powerset *}
    4.60 -
    4.61 -lemma cantor1: "\<not> (\<exists>f:: 'a \<Rightarrow> 'a set. \<forall>S. \<exists>x. f x = S)"
    4.62 -  -- {* Requires best-first search because it is undirectional. *}
    4.63 -  by best
    4.64 -
    4.65 -schematic_lemma "\<forall>f:: 'a \<Rightarrow> 'a set. \<forall>x. f x \<noteq> ?S f"
    4.66 -  -- {*This form displays the diagonal term. *}
    4.67 -  by best
    4.68 -
    4.69 -schematic_lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
    4.70 -  -- {* This form exploits the set constructs. *}
    4.71 -  by (rule notI, erule rangeE, best)
    4.72 -
    4.73 -schematic_lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
    4.74 -  -- {* Or just this! *}
    4.75 -  by best
    4.76 -
    4.77 -
    4.78 -subsection {* The Schröder-Berstein Theorem *}
    4.79 -
    4.80 -lemma disj_lemma: "- (f ` X) = g ` (-X) \<Longrightarrow> f a = g b \<Longrightarrow> a \<in> X \<Longrightarrow> b \<in> X"
    4.81 -  by blast
    4.82 -
    4.83 -lemma surj_if_then_else:
    4.84 -  "-(f ` X) = g ` (-X) \<Longrightarrow> surj (\<lambda>z. if z \<in> X then f z else g z)"
    4.85 -  by (simp add: surj_def) blast
    4.86 -
    4.87 -lemma bij_if_then_else:
    4.88 -  "inj_on f X \<Longrightarrow> inj_on g (-X) \<Longrightarrow> -(f ` X) = g ` (-X) \<Longrightarrow>
    4.89 -    h = (\<lambda>z. if z \<in> X then f z else g z) \<Longrightarrow> inj h \<and> surj h"
    4.90 -  apply (unfold inj_on_def)
    4.91 -  apply (simp add: surj_if_then_else)
    4.92 -  apply (blast dest: disj_lemma sym)
    4.93 -  done
    4.94 -
    4.95 -lemma decomposition: "\<exists>X. X = - (g ` (- (f ` X)))"
    4.96 -  apply (rule exI)
    4.97 -  apply (rule lfp_unfold)
    4.98 -  apply (rule monoI, blast)
    4.99 -  done
   4.100 -
   4.101 -theorem Schroeder_Bernstein:
   4.102 -  "inj (f :: 'a \<Rightarrow> 'b) \<Longrightarrow> inj (g :: 'b \<Rightarrow> 'a)
   4.103 -    \<Longrightarrow> \<exists>h:: 'a \<Rightarrow> 'b. inj h \<and> surj h"
   4.104 -  apply (rule decomposition [where f=f and g=g, THEN exE])
   4.105 -  apply (rule_tac x = "(\<lambda>z. if z \<in> x then f z else inv g z)" in exI) 
   4.106 -    --{*The term above can be synthesized by a sufficiently detailed proof.*}
   4.107 -  apply (rule bij_if_then_else)
   4.108 -     apply (rule_tac [4] refl)
   4.109 -    apply (rule_tac [2] inj_on_inv_into)
   4.110 -    apply (erule subset_inj_on [OF _ subset_UNIV])
   4.111 -   apply blast
   4.112 -  apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric])
   4.113 -  done
   4.114 -
   4.115 -
   4.116 -subsection {* A simple party theorem *}
   4.117 -
   4.118 -text{* \emph{At any party there are two people who know the same
   4.119 -number of people}. Provided the party consists of at least two people
   4.120 -and the knows relation is symmetric. Knowing yourself does not count
   4.121 ---- otherwise knows needs to be reflexive. (From Freek Wiedijk's talk
   4.122 -at TPHOLs 2007.) *}
   4.123 -
   4.124 -lemma equal_number_of_acquaintances:
   4.125 -assumes "Domain R <= A" and "sym R" and "card A \<ge> 2"
   4.126 -shows "\<not> inj_on (%a. card(R `` {a} - {a})) A"
   4.127 -proof -
   4.128 -  let ?N = "%a. card(R `` {a} - {a})"
   4.129 -  let ?n = "card A"
   4.130 -  have "finite A" using `card A \<ge> 2` by(auto intro:ccontr)
   4.131 -  have 0: "R `` A <= A" using `sym R` `Domain R <= A`
   4.132 -    unfolding Domain_def sym_def by blast
   4.133 -  have h: "ALL a:A. R `` {a} <= A" using 0 by blast
   4.134 -  hence 1: "ALL a:A. finite(R `` {a})" using `finite A`
   4.135 -    by(blast intro: finite_subset)
   4.136 -  have sub: "?N ` A <= {0..<?n}"
   4.137 -  proof -
   4.138 -    have "ALL a:A. R `` {a} - {a} < A" using h by blast
   4.139 -    thus ?thesis using psubset_card_mono[OF `finite A`] by auto
   4.140 -  qed
   4.141 -  show "~ inj_on ?N A" (is "~ ?I")
   4.142 -  proof
   4.143 -    assume ?I
   4.144 -    hence "?n = card(?N ` A)" by(rule card_image[symmetric])
   4.145 -    with sub `finite A` have 2[simp]: "?N ` A = {0..<?n}"
   4.146 -      using subset_card_intvl_is_intvl[of _ 0] by(auto)
   4.147 -    have "0 : ?N ` A" and "?n - 1 : ?N ` A"  using `card A \<ge> 2` by simp+
   4.148 -    then obtain a b where ab: "a:A" "b:A" and Na: "?N a = 0" and Nb: "?N b = ?n - 1"
   4.149 -      by (auto simp del: 2)
   4.150 -    have "a \<noteq> b" using Na Nb `card A \<ge> 2` by auto
   4.151 -    have "R `` {a} - {a} = {}" by (metis 1 Na ab card_eq_0_iff finite_Diff)
   4.152 -    hence "b \<notin> R `` {a}" using `a\<noteq>b` by blast
   4.153 -    hence "a \<notin> R `` {b}" by (metis Image_singleton_iff assms(2) sym_def)
   4.154 -    hence 3: "R `` {b} - {b} <= A - {a,b}" using 0 ab by blast
   4.155 -    have 4: "finite (A - {a,b})" using `finite A` by simp
   4.156 -    have "?N b <= ?n - 2" using ab `a\<noteq>b` `finite A` card_mono[OF 4 3] by simp
   4.157 -    then show False using Nb `card A \<ge>  2` by arith
   4.158 -  qed
   4.159 -qed
   4.160 -
   4.161 -text {*
   4.162 -  From W. W. Bledsoe and Guohui Feng, SET-VAR. JAR 11 (3), 1993, pages
   4.163 -  293-314.
   4.164 -
   4.165 -  Isabelle can prove the easy examples without any special mechanisms,
   4.166 -  but it can't prove the hard ones.
   4.167 -*}
   4.168 -
   4.169 -lemma "\<exists>A. (\<forall>x \<in> A. x \<le> (0::int))"
   4.170 -  -- {* Example 1, page 295. *}
   4.171 -  by force
   4.172 -
   4.173 -lemma "D \<in> F \<Longrightarrow> \<exists>G. \<forall>A \<in> G. \<exists>B \<in> F. A \<subseteq> B"
   4.174 -  -- {* Example 2. *}
   4.175 -  by force
   4.176 -
   4.177 -lemma "P a \<Longrightarrow> \<exists>A. (\<forall>x \<in> A. P x) \<and> (\<exists>y. y \<in> A)"
   4.178 -  -- {* Example 3. *}
   4.179 -  by force
   4.180 -
   4.181 -lemma "a < b \<and> b < (c::int) \<Longrightarrow> \<exists>A. a \<notin> A \<and> b \<in> A \<and> c \<notin> A"
   4.182 -  -- {* Example 4. *}
   4.183 -  by force
   4.184 -
   4.185 -lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
   4.186 -  -- {*Example 5, page 298. *}
   4.187 -  by force
   4.188 -
   4.189 -lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
   4.190 -  -- {* Example 6. *}
   4.191 -  by force
   4.192 -
   4.193 -lemma "\<exists>A. a \<notin> A"
   4.194 -  -- {* Example 7. *}
   4.195 -  by force
   4.196 -
   4.197 -lemma "(\<forall>u v. u < (0::int) \<longrightarrow> u \<noteq> abs v)
   4.198 -    \<longrightarrow> (\<exists>A::int set. (\<forall>y. abs y \<notin> A) \<and> -2 \<in> A)"
   4.199 -  -- {* Example 8 now needs a small hint. *}
   4.200 -  by (simp add: abs_if, force)
   4.201 -    -- {* not @{text blast}, which can't simplify @{text "-2 < 0"} *}
   4.202 -
   4.203 -text {* Example 9 omitted (requires the reals). *}
   4.204 -
   4.205 -text {* The paper has no Example 10! *}
   4.206 -
   4.207 -lemma "(\<forall>A. 0 \<in> A \<and> (\<forall>x \<in> A. Suc x \<in> A) \<longrightarrow> n \<in> A) \<and>
   4.208 -  P 0 \<and> (\<forall>x. P x \<longrightarrow> P (Suc x)) \<longrightarrow> P n"
   4.209 -  -- {* Example 11: needs a hint. *}
   4.210 -by(metis nat.induct)
   4.211 -
   4.212 -lemma
   4.213 -  "(\<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)
   4.214 -    \<and> P n \<longrightarrow> P m"
   4.215 -  -- {* Example 12. *}
   4.216 -  by auto
   4.217 -
   4.218 -lemma
   4.219 -  "(\<forall>x. (\<exists>u. x = 2 * u) = (\<not> (\<exists>v. Suc x = 2 * v))) \<longrightarrow>
   4.220 -    (\<exists>A. \<forall>x. (x \<in> A) = (Suc x \<notin> A))"
   4.221 -  -- {* Example EO1: typo in article, and with the obvious fix it seems
   4.222 -      to require arithmetic reasoning. *}
   4.223 -  apply clarify
   4.224 -  apply (rule_tac x = "{x. \<exists>u. x = 2 * u}" in exI, auto)
   4.225 -   apply metis+
   4.226 -  done
   4.227 -
   4.228 -end