author | ballarin |
Tue, 04 Jul 2006 11:36:08 +0200 | |
changeset 19982 | e4d50f8f3722 |
parent 18391 | 2e901da7cd3a |
child 24573 | 5bbdc9b60648 |
permissions | -rw-r--r-- |
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(* Title: HOL/ex/set.thy |
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ID: $Id$ |
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Author: Tobias Nipkow and Lawrence C Paulson |
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Copyright 1991 University of Cambridge |
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*) |
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header {* Set Theory examples: Cantor's Theorem, Schröder-Bernstein Theorem, etc. *} |
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theory set imports Main begin |
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text{* |
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These two are cited in Benzmueller and Kohlhase's system description |
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of LEO, CADE-15, 1998 (pages 139-143) as theorems LEO could not |
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prove. |
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*} |
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lemma "(X = Y \<union> Z) = |
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(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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by blast |
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lemma "(X = Y \<inter> Z) = |
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(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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by blast |
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text {* |
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Trivial example of term synthesis: apparently hard for some provers! |
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*} |
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lemma "a \<noteq> b \<Longrightarrow> a \<in> ?X \<and> b \<notin> ?X" |
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by blast |
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subsection {* Examples for the @{text blast} paper *} |
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lemma "(\<Union>x \<in> C. f x \<union> g x) = \<Union>(f ` C) \<union> \<Union>(g ` C)" |
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-- {* Union-image, called @{text Un_Union_image} in Main HOL *} |
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by blast |
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lemma "(\<Inter>x \<in> C. f x \<inter> g x) = \<Inter>(f ` C) \<inter> \<Inter>(g ` C)" |
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-- {* Inter-image, called @{text Int_Inter_image} in Main HOL *} |
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by blast |
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lemma singleton_example_1: |
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"\<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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by blast |
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(*With removal of negated equality literals, this no longer works: |
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by (meson subsetI subset_antisym insertCI) |
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*) |
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lemma singleton_example_2: |
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"\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}" |
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-- {*Variant of the problem above. *} |
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by blast |
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(*With removal of negated equality literals, this no longer works: |
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by (meson subsetI subset_antisym insertCI UnionI) |
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*) |
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lemma "\<exists>!x. f (g x) = x \<Longrightarrow> \<exists>!y. g (f y) = y" |
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-- {* A unique fixpoint theorem --- @{text fast}/@{text best}/@{text meson} all fail. *} |
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apply (erule ex1E, rule ex1I, erule arg_cong) |
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apply (rule subst, assumption, erule allE, rule arg_cong, erule mp) |
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apply (erule arg_cong) |
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done |
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subsection {* Cantor's Theorem: There is no surjection from a set to its powerset *} |
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lemma cantor1: "\<not> (\<exists>f:: 'a \<Rightarrow> 'a set. \<forall>S. \<exists>x. f x = S)" |
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-- {* Requires best-first search because it is undirectional. *} |
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by best |
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lemma "\<forall>f:: 'a \<Rightarrow> 'a set. \<forall>x. f x \<noteq> ?S f" |
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-- {*This form displays the diagonal term. *} |
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by best |
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lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)" |
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-- {* This form exploits the set constructs. *} |
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by (rule notI, erule rangeE, best) |
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lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)" |
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-- {* Or just this! *} |
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by best |
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subsection {* The Schröder-Berstein Theorem *} |
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lemma disj_lemma: "- (f ` X) = g ` (-X) \<Longrightarrow> f a = g b \<Longrightarrow> a \<in> X \<Longrightarrow> b \<in> X" |
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by blast |
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lemma surj_if_then_else: |
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"-(f ` X) = g ` (-X) \<Longrightarrow> surj (\<lambda>z. if z \<in> X then f z else g z)" |
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by (simp add: surj_def) blast |
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lemma bij_if_then_else: |
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"inj_on f X \<Longrightarrow> inj_on g (-X) \<Longrightarrow> -(f ` X) = g ` (-X) \<Longrightarrow> |
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h = (\<lambda>z. if z \<in> X then f z else g z) \<Longrightarrow> inj h \<and> surj h" |
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apply (unfold inj_on_def) |
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apply (simp add: surj_if_then_else) |
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apply (blast dest: disj_lemma sym) |
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done |
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lemma decomposition: "\<exists>X. X = - (g ` (- (f ` X)))" |
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apply (rule exI) |
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apply (rule lfp_unfold) |
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apply (rule monoI, blast) |
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done |
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theorem Schroeder_Bernstein: |
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"inj (f :: 'a \<Rightarrow> 'b) \<Longrightarrow> inj (g :: 'b \<Rightarrow> 'a) |
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\<Longrightarrow> \<exists>h:: 'a \<Rightarrow> 'b. inj h \<and> surj h" |
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apply (rule decomposition [where f=f and g=g, THEN exE]) |
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apply (rule_tac x = "(\<lambda>z. if z \<in> x then f z else inv g z)" in exI) |
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--{*The term above can be synthesized by a sufficiently detailed proof.*} |
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apply (rule bij_if_then_else) |
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apply (rule_tac [4] refl) |
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apply (rule_tac [2] inj_on_inv) |
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apply (erule subset_inj_on [OF _ subset_UNIV]) |
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apply blast |
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apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric]) |
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done |
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text {* |
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From W. W. Bledsoe and Guohui Feng, SET-VAR. JAR 11 (3), 1993, pages |
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293-314. |
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Isabelle can prove the easy examples without any special mechanisms, |
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but it can't prove the hard ones. |
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*} |
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lemma "\<exists>A. (\<forall>x \<in> A. x \<le> (0::int))" |
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-- {* Example 1, page 295. *} |
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by force |
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lemma "D \<in> F \<Longrightarrow> \<exists>G. \<forall>A \<in> G. \<exists>B \<in> F. A \<subseteq> B" |
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-- {* Example 2. *} |
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by force |
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lemma "P a \<Longrightarrow> \<exists>A. (\<forall>x \<in> A. P x) \<and> (\<exists>y. y \<in> A)" |
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-- {* Example 3. *} |
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by force |
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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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-- {* Example 4. *} |
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by force |
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lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A" |
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-- {*Example 5, page 298. *} |
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by force |
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lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A" |
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-- {* Example 6. *} |
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by force |
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lemma "\<exists>A. a \<notin> A" |
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-- {* Example 7. *} |
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by force |
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lemma "(\<forall>u v. u < (0::int) \<longrightarrow> u \<noteq> abs v) |
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\<longrightarrow> (\<exists>A::int set. (\<forall>y. abs y \<notin> A) \<and> -2 \<in> A)" |
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-- {* Example 8 now needs a small hint. *} |
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by (simp add: abs_if, force) |
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-- {* not @{text blast}, which can't simplify @{text "-2 < 0"} *} |
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text {* Example 9 omitted (requires the reals). *} |
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text {* The paper has no Example 10! *} |
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lemma "(\<forall>A. 0 \<in> A \<and> (\<forall>x \<in> A. Suc x \<in> A) \<longrightarrow> n \<in> A) \<and> |
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P 0 \<and> (\<forall>x. P x \<longrightarrow> P (Suc x)) \<longrightarrow> P n" |
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-- {* Example 11: needs a hint. *} |
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apply clarify |
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apply (drule_tac x = "{x. P x}" in spec) |
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apply force |
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done |
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lemma |
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"(\<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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\<and> P n \<longrightarrow> P m" |
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-- {* Example 12. *} |
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by auto |
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lemma |
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"(\<forall>x. (\<exists>u. x = 2 * u) = (\<not> (\<exists>v. Suc x = 2 * v))) \<longrightarrow> |
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(\<exists>A. \<forall>x. (x \<in> A) = (Suc x \<notin> A))" |
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-- {* Example EO1: typo in article, and with the obvious fix it seems |
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to require arithmetic reasoning. *} |
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apply clarify |
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apply (rule_tac x = "{x. \<exists>u. x = 2 * u}" in exI, auto) |
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apply (case_tac v, auto) |
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apply (drule_tac x = "Suc v" and P = "\<lambda>x. ?a x \<noteq> ?b x" in spec, force) |
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done |
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end |