src/HOL/Hilbert_Choice.thy
author wenzelm
Mon Aug 08 19:34:00 2016 +0200 (2016-08-08)
changeset 63630 b2a6a1a49d39
parent 63629 3b3ab4674274
child 63807 5f77017055a3
permissions -rw-r--r--
tuned proof;
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(*  Title:      HOL/Hilbert_Choice.thy
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    Author:     Lawrence C Paulson, Tobias Nipkow
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    Copyright   2001  University of Cambridge
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*)
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section \<open>Hilbert's Epsilon-Operator and the Axiom of Choice\<close>
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theory Hilbert_Choice
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  imports Wellfounded
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  keywords "specification" :: thy_goal
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begin
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subsection \<open>Hilbert's epsilon\<close>
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axiomatization Eps :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
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  where someI: "P x \<Longrightarrow> P (Eps P)"
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syntax (epsilon)
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  "_Eps" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a"  ("(3\<some>_./ _)" [0, 10] 10)
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syntax (input)
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  "_Eps" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a"  ("(3@ _./ _)" [0, 10] 10)
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syntax
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  "_Eps" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a"  ("(3SOME _./ _)" [0, 10] 10)
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translations
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  "SOME x. P" \<rightleftharpoons> "CONST Eps (\<lambda>x. P)"
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print_translation \<open>
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  [(@{const_syntax Eps}, fn _ => fn [Abs abs] =>
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      let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
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      in Syntax.const @{syntax_const "_Eps"} $ x $ t end)]
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\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
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definition inv_into :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"
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  where "inv_into A f \<equiv> \<lambda>x. SOME y. y \<in> A \<and> f y = x"
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abbreviation inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"
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  where "inv \<equiv> inv_into UNIV"
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subsection \<open>Hilbert's Epsilon-operator\<close>
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text \<open>
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  Easier to apply than \<open>someI\<close> if the witness comes from an
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  existential formula.
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\<close>
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lemma someI_ex [elim?]: "\<exists>x. P x \<Longrightarrow> P (SOME x. P x)"
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  apply (erule exE)
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  apply (erule someI)
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  done
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text \<open>
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  Easier to apply than \<open>someI\<close> because the conclusion has only one
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  occurrence of @{term P}.
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\<close>
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lemma someI2: "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (SOME x. P x)"
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  by (blast intro: someI)
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text \<open>
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  Easier to apply than \<open>someI2\<close> if the witness comes from an
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  existential formula.
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\<close>
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lemma someI2_ex: "\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (SOME x. P x)"
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  by (blast intro: someI2)
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lemma someI2_bex: "\<exists>a\<in>A. P a \<Longrightarrow> (\<And>x. x \<in> A \<and> P x \<Longrightarrow> Q x) \<Longrightarrow> Q (SOME x. x \<in> A \<and> P x)"
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  by (blast intro: someI2)
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lemma some_equality [intro]: "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> x = a) \<Longrightarrow> (SOME x. P x) = a"
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  by (blast intro: someI2)
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lemma some1_equality: "\<exists>!x. P x \<Longrightarrow> P a \<Longrightarrow> (SOME x. P x) = a"
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  by blast
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lemma some_eq_ex: "P (SOME x. P x) \<longleftrightarrow> (\<exists>x. P x)"
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  by (blast intro: someI)
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lemma some_in_eq: "(SOME x. x \<in> A) \<in> A \<longleftrightarrow> A \<noteq> {}"
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  unfolding ex_in_conv[symmetric] by (rule some_eq_ex)
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lemma some_eq_trivial [simp]: "(SOME y. y = x) = x"
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  by (rule some_equality) (rule refl)
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lemma some_sym_eq_trivial [simp]: "(SOME y. x = y) = x"
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  apply (rule some_equality)
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   apply (rule refl)
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  apply (erule sym)
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  done
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subsection \<open>Axiom of Choice, Proved Using the Description Operator\<close>
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lemma choice: "\<forall>x. \<exists>y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)"
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  by (fast elim: someI)
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lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y \<Longrightarrow> \<exists>f. \<forall>x\<in>S. Q x (f x)"
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  by (fast elim: someI)
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lemma choice_iff: "(\<forall>x. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x. Q x (f x))"
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  by (fast elim: someI)
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lemma choice_iff': "(\<forall>x. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x. P x \<longrightarrow> Q x (f x))"
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  by (fast elim: someI)
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lemma bchoice_iff: "(\<forall>x\<in>S. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. Q x (f x))"
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  by (fast elim: someI)
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lemma bchoice_iff': "(\<forall>x\<in>S. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. P x \<longrightarrow> Q x (f x))"
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  by (fast elim: someI)
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lemma dependent_nat_choice:
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  assumes 1: "\<exists>x. P 0 x"
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    and 2: "\<And>x n. P n x \<Longrightarrow> \<exists>y. P (Suc n) y \<and> Q n x y"
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  shows "\<exists>f. \<forall>n. P n (f n) \<and> Q n (f n) (f (Suc n))"
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proof (intro exI allI conjI)
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  fix n
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  define f where "f = rec_nat (SOME x. P 0 x) (\<lambda>n x. SOME y. P (Suc n) y \<and> Q n x y)"
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  then have "P 0 (f 0)" "\<And>n. P n (f n) \<Longrightarrow> P (Suc n) (f (Suc n)) \<and> Q n (f n) (f (Suc n))"
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    using someI_ex[OF 1] someI_ex[OF 2] by simp_all
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  then show "P n (f n)" "Q n (f n) (f (Suc n))"
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    by (induct n) auto
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qed
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subsection \<open>Function Inverse\<close>
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lemma inv_def: "inv f = (\<lambda>y. SOME x. f x = y)"
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  by (simp add: inv_into_def)
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lemma inv_into_into: "x \<in> f ` A \<Longrightarrow> inv_into A f x \<in> A"
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  by (simp add: inv_into_def) (fast intro: someI2)
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lemma inv_identity [simp]: "inv (\<lambda>a. a) = (\<lambda>a. a)"
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  by (simp add: inv_def)
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lemma inv_id [simp]: "inv id = id"
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  by (simp add: id_def)
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lemma inv_into_f_f [simp]: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> inv_into A f (f x) = x"
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  by (simp add: inv_into_def inj_on_def) (blast intro: someI2)
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lemma inv_f_f: "inj f \<Longrightarrow> inv f (f x) = x"
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  by simp
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lemma f_inv_into_f: "y : f`A \<Longrightarrow> f (inv_into A f y) = y"
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  by (simp add: inv_into_def) (fast intro: someI2)
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lemma inv_into_f_eq: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> f x = y \<Longrightarrow> inv_into A f y = x"
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  by (erule subst) (fast intro: inv_into_f_f)
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lemma inv_f_eq: "inj f \<Longrightarrow> f x = y \<Longrightarrow> inv f y = x"
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  by (simp add:inv_into_f_eq)
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lemma inj_imp_inv_eq: "inj f \<Longrightarrow> \<forall>x. f (g x) = x \<Longrightarrow> inv f = g"
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  by (blast intro: inv_into_f_eq)
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text \<open>But is it useful?\<close>
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lemma inj_transfer:
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  assumes inj: "inj f"
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    and minor: "\<And>y. y \<in> range f \<Longrightarrow> P (inv f y)"
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  shows "P x"
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proof -
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  have "f x \<in> range f" by auto
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  then have "P(inv f (f x))" by (rule minor)
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  then show "P x" by (simp add: inv_into_f_f [OF inj])
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qed
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lemma inj_iff: "inj f \<longleftrightarrow> inv f \<circ> f = id"
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  by (simp add: o_def fun_eq_iff) (blast intro: inj_on_inverseI inv_into_f_f)
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lemma inv_o_cancel[simp]: "inj f \<Longrightarrow> inv f \<circ> f = id"
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  by (simp add: inj_iff)
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lemma o_inv_o_cancel[simp]: "inj f \<Longrightarrow> g \<circ> inv f \<circ> f = g"
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  by (simp add: comp_assoc)
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lemma inv_into_image_cancel[simp]: "inj_on f A \<Longrightarrow> S \<subseteq> A \<Longrightarrow> inv_into A f ` f ` S = S"
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  by (fastforce simp: image_def)
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lemma inj_imp_surj_inv: "inj f \<Longrightarrow> surj (inv f)"
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  by (blast intro!: surjI inv_into_f_f)
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lemma surj_f_inv_f: "surj f \<Longrightarrow> f (inv f y) = y"
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  by (simp add: f_inv_into_f)
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lemma inv_into_injective:
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  assumes eq: "inv_into A f x = inv_into A f y"
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    and x: "x \<in> f`A"
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    and y: "y \<in> f`A"
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  shows "x = y"
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proof -
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  from eq have "f (inv_into A f x) = f (inv_into A f y)"
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    by simp
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  with x y show ?thesis
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    by (simp add: f_inv_into_f)
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qed
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lemma inj_on_inv_into: "B \<subseteq> f`A \<Longrightarrow> inj_on (inv_into A f) B"
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  by (blast intro: inj_onI dest: inv_into_injective injD)
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lemma bij_betw_inv_into: "bij_betw f A B \<Longrightarrow> bij_betw (inv_into A f) B A"
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  by (auto simp add: bij_betw_def inj_on_inv_into)
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lemma surj_imp_inj_inv: "surj f \<Longrightarrow> inj (inv f)"
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  by (simp add: inj_on_inv_into)
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lemma surj_iff: "surj f \<longleftrightarrow> f \<circ> inv f = id"
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  by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
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lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
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  by (simp add: o_def surj_iff fun_eq_iff)
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lemma surj_imp_inv_eq: "surj f \<Longrightarrow> \<forall>x. g (f x) = x \<Longrightarrow> inv f = g"
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  apply (rule ext)
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  apply (drule_tac x = "inv f x" in spec)
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  apply (simp add: surj_f_inv_f)
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  done
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lemma bij_imp_bij_inv: "bij f \<Longrightarrow> bij (inv f)"
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  by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
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lemma inv_equality: "(\<And>x. g (f x) = x) \<Longrightarrow> (\<And>y. f (g y) = y) \<Longrightarrow> inv f = g"
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  by (rule ext) (auto simp add: inv_into_def)
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lemma inv_inv_eq: "bij f \<Longrightarrow> inv (inv f) = f"
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  by (rule inv_equality) (auto simp add: bij_def surj_f_inv_f)
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text \<open>
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  \<open>bij (inv f)\<close> implies little about \<open>f\<close>. Consider \<open>f :: bool \<Rightarrow> bool\<close> such
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  that \<open>f True = f False = True\<close>. Then it ia consistent with axiom \<open>someI\<close>
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  that \<open>inv f\<close> could be any function at all, including the identity function.
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  If \<open>inv f = id\<close> then \<open>inv f\<close> is a bijection, but \<open>inj f\<close>, \<open>surj f\<close> and \<open>inv
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  (inv f) = f\<close> all fail.
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\<close>
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lemma inv_into_comp:
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  "inj_on f (g ` A) \<Longrightarrow> inj_on g A \<Longrightarrow> x \<in> f ` g ` A \<Longrightarrow>
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    inv_into A (f \<circ> g) x = (inv_into A g \<circ> inv_into (g ` A) f) x"
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  apply (rule inv_into_f_eq)
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    apply (fast intro: comp_inj_on)
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   apply (simp add: inv_into_into)
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  apply (simp add: f_inv_into_f inv_into_into)
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  done
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lemma o_inv_distrib: "bij f \<Longrightarrow> bij g \<Longrightarrow> inv (f \<circ> g) = inv g \<circ> inv f"
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  by (rule inv_equality) (auto simp add: bij_def surj_f_inv_f)
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lemma image_surj_f_inv_f: "surj f \<Longrightarrow> f ` (inv f ` A) = A"
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  by (simp add: surj_f_inv_f image_comp comp_def)
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lemma image_inv_f_f: "inj f \<Longrightarrow> inv f ` (f ` A) = A"
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  by simp
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lemma inv_image_comp: "inj f \<Longrightarrow> inv f ` (f ` X) = X"
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  by (fact image_inv_f_f)
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lemma bij_image_Collect_eq: "bij f \<Longrightarrow> f ` Collect P = {y. P (inv f y)}"
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  apply auto
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   apply (force simp add: bij_is_inj)
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  apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
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  done
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lemma bij_vimage_eq_inv_image: "bij f \<Longrightarrow> f -` A = inv f ` A"
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  apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
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  apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
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  done
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lemma finite_fun_UNIVD1:
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  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
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    and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
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  shows "finite (UNIV :: 'a set)"
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proof -
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  let ?UNIV_b = "UNIV :: 'b set"
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  from fin have "finite ?UNIV_b"
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    by (rule finite_fun_UNIVD2)
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  with card have "card ?UNIV_b \<ge> Suc (Suc 0)"
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    by (cases "card ?UNIV_b") (auto simp: card_eq_0_iff)
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  then have "card ?UNIV_b = Suc (Suc (card ?UNIV_b - Suc (Suc 0)))"
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    by simp
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  then obtain b1 b2 :: 'b where b1b2: "b1 \<noteq> b2"
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    by (auto simp: card_Suc_eq)
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  from fin have fin': "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))"
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    by (rule finite_imageI)
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  have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
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  proof (rule UNIV_eq_I)
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    fix x :: 'a
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    from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1"
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      by (simp add: inv_into_def)
wenzelm@63612
   288
    then show "x \<in> range (\<lambda>f::'a \<Rightarrow> 'b. inv f b1)"
wenzelm@63612
   289
      by blast
haftmann@31380
   290
  qed
wenzelm@63630
   291
  with fin' show ?thesis
wenzelm@63612
   292
    by simp
haftmann@31380
   293
qed
paulson@14760
   294
wenzelm@60758
   295
text \<open>
traytel@54578
   296
  Every infinite set contains a countable subset. More precisely we
wenzelm@61799
   297
  show that a set \<open>S\<close> is infinite if and only if there exists an
wenzelm@61799
   298
  injective function from the naturals into \<open>S\<close>.
traytel@54578
   299
traytel@54578
   300
  The ``only if'' direction is harder because it requires the
traytel@54578
   301
  construction of a sequence of pairwise different elements of an
wenzelm@61799
   302
  infinite set \<open>S\<close>. The idea is to construct a sequence of
wenzelm@61799
   303
  non-empty and infinite subsets of \<open>S\<close> obtained by successively
wenzelm@61799
   304
  removing elements of \<open>S\<close>.
wenzelm@60758
   305
\<close>
traytel@54578
   306
traytel@54578
   307
lemma infinite_countable_subset:
wenzelm@63629
   308
  assumes inf: "\<not> finite S"
wenzelm@63629
   309
  shows "\<exists>f::nat \<Rightarrow> 'a. inj f \<and> range f \<subseteq> S"
wenzelm@61799
   310
  \<comment> \<open>Courtesy of Stephan Merz\<close>
traytel@54578
   311
proof -
wenzelm@63040
   312
  define Sseq where "Sseq = rec_nat S (\<lambda>n T. T - {SOME e. e \<in> T})"
wenzelm@63040
   313
  define pick where "pick n = (SOME e. e \<in> Sseq n)" for n
wenzelm@63540
   314
  have *: "Sseq n \<subseteq> S" "\<not> finite (Sseq n)" for n
wenzelm@63612
   315
    by (induct n) (auto simp: Sseq_def inf)
wenzelm@63540
   316
  then have **: "\<And>n. pick n \<in> Sseq n"
traytel@55811
   317
    unfolding pick_def by (subst (asm) finite.simps) (auto simp add: ex_in_conv intro: someI_ex)
wenzelm@63540
   318
  with * have "range pick \<subseteq> S" by auto
wenzelm@63612
   319
  moreover have "pick n \<noteq> pick (n + Suc m)" for m n
wenzelm@63612
   320
  proof -
wenzelm@63540
   321
    have "pick n \<notin> Sseq (n + Suc m)"
wenzelm@63540
   322
      by (induct m) (auto simp add: Sseq_def pick_def)
wenzelm@63612
   323
    with ** show ?thesis by auto
wenzelm@63612
   324
  qed
wenzelm@63612
   325
  then have "inj pick"
wenzelm@63612
   326
    by (intro linorder_injI) (auto simp add: less_iff_Suc_add)
traytel@54578
   327
  ultimately show ?thesis by blast
traytel@54578
   328
qed
traytel@54578
   329
wenzelm@63629
   330
lemma infinite_iff_countable_subset: "\<not> finite S \<longleftrightarrow> (\<exists>f::nat \<Rightarrow> 'a. inj f \<and> range f \<subseteq> S)"
wenzelm@61799
   331
  \<comment> \<open>Courtesy of Stephan Merz\<close>
traytel@55811
   332
  using finite_imageD finite_subset infinite_UNIV_char_0 infinite_countable_subset by auto
traytel@54578
   333
hoelzl@40703
   334
lemma image_inv_into_cancel:
wenzelm@63612
   335
  assumes surj: "f`A = A'"
wenzelm@63612
   336
    and sub: "B' \<subseteq> A'"
hoelzl@40703
   337
  shows "f `((inv_into A f)`B') = B'"
hoelzl@40703
   338
  using assms
wenzelm@63612
   339
proof (auto simp: f_inv_into_f)
wenzelm@63612
   340
  let ?f' = "inv_into A f"
wenzelm@63612
   341
  fix a'
wenzelm@63612
   342
  assume *: "a' \<in> B'"
wenzelm@63612
   343
  with sub have "a' \<in> A'" by auto
wenzelm@63612
   344
  with surj have "a' = f (?f' a')"
wenzelm@63612
   345
    by (auto simp: f_inv_into_f)
wenzelm@63612
   346
  with * show "a' \<in> f ` (?f' ` B')" by blast
hoelzl@40703
   347
qed
hoelzl@40703
   348
hoelzl@40703
   349
lemma inv_into_inv_into_eq:
wenzelm@63612
   350
  assumes "bij_betw f A A'"
wenzelm@63612
   351
    and a: "a \<in> A"
hoelzl@40703
   352
  shows "inv_into A' (inv_into A f) a = f a"
hoelzl@40703
   353
proof -
wenzelm@63612
   354
  let ?f' = "inv_into A f"
wenzelm@63612
   355
  let ?f'' = "inv_into A' ?f'"
wenzelm@63612
   356
  from assms have *: "bij_betw ?f' A' A"
wenzelm@63612
   357
    by (auto simp: bij_betw_inv_into)
wenzelm@63612
   358
  with a obtain a' where a': "a' \<in> A'" "?f' a' = a"
wenzelm@63612
   359
    unfolding bij_betw_def by force
wenzelm@63612
   360
  with a * have "?f'' a = a'"
wenzelm@63612
   361
    by (auto simp: f_inv_into_f bij_betw_def)
wenzelm@63612
   362
  moreover from assms a' have "f a = a'"
wenzelm@63612
   363
    by (auto simp: bij_betw_def)
hoelzl@40703
   364
  ultimately show "?f'' a = f a" by simp
hoelzl@40703
   365
qed
hoelzl@40703
   366
hoelzl@40703
   367
lemma inj_on_iff_surj:
hoelzl@40703
   368
  assumes "A \<noteq> {}"
wenzelm@63629
   369
  shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
hoelzl@40703
   370
proof safe
wenzelm@63612
   371
  fix f
wenzelm@63612
   372
  assume inj: "inj_on f A" and incl: "f ` A \<subseteq> A'"
wenzelm@63612
   373
  let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"
wenzelm@63612
   374
  let ?csi = "\<lambda>a. a \<in> A"
hoelzl@40703
   375
  let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
hoelzl@40703
   376
  have "?g ` A' = A"
hoelzl@40703
   377
  proof
wenzelm@63612
   378
    show "?g ` A' \<subseteq> A"
hoelzl@40703
   379
    proof clarify
wenzelm@63612
   380
      fix a'
wenzelm@63612
   381
      assume *: "a' \<in> A'"
hoelzl@40703
   382
      show "?g a' \<in> A"
wenzelm@63612
   383
      proof (cases "a' \<in> f ` A")
wenzelm@63612
   384
        case True
hoelzl@40703
   385
        then obtain a where "?phi a' a" by blast
wenzelm@63612
   386
        then have "?phi a' (SOME a. ?phi a' a)"
wenzelm@63612
   387
          using someI[of "?phi a'" a] by blast
wenzelm@63612
   388
        with True show ?thesis by auto
hoelzl@40703
   389
      next
wenzelm@63612
   390
        case False
wenzelm@63612
   391
        with assms have "?csi (SOME a. ?csi a)"
wenzelm@63612
   392
          using someI_ex[of ?csi] by blast
wenzelm@63612
   393
        with False show ?thesis by auto
hoelzl@40703
   394
      qed
hoelzl@40703
   395
    qed
hoelzl@40703
   396
  next
wenzelm@63612
   397
    show "A \<subseteq> ?g ` A'"
wenzelm@63612
   398
    proof -
wenzelm@63612
   399
      have "?g (f a) = a \<and> f a \<in> A'" if a: "a \<in> A" for a
wenzelm@63612
   400
      proof -
wenzelm@63612
   401
        let ?b = "SOME aa. ?phi (f a) aa"
wenzelm@63612
   402
        from a have "?phi (f a) a" by auto
wenzelm@63612
   403
        then have *: "?phi (f a) ?b"
wenzelm@63612
   404
          using someI[of "?phi(f a)" a] by blast
wenzelm@63612
   405
        then have "?g (f a) = ?b" using a by auto
wenzelm@63612
   406
        moreover from inj * a have "a = ?b"
wenzelm@63612
   407
          by (auto simp add: inj_on_def)
wenzelm@63612
   408
        ultimately have "?g(f a) = a" by simp
wenzelm@63612
   409
        with incl a show ?thesis by auto
wenzelm@63612
   410
      qed
wenzelm@63612
   411
      then show ?thesis by force
hoelzl@40703
   412
    qed
hoelzl@40703
   413
  qed
wenzelm@63612
   414
  then show "\<exists>g. g ` A' = A" by blast
hoelzl@40703
   415
next
wenzelm@63612
   416
  fix g
wenzelm@63612
   417
  let ?f = "inv_into A' g"
hoelzl@40703
   418
  have "inj_on ?f (g ` A')"
wenzelm@63612
   419
    by (auto simp: inj_on_inv_into)
wenzelm@63612
   420
  moreover have "?f (g a') \<in> A'" if a': "a' \<in> A'" for a'
wenzelm@63612
   421
  proof -
wenzelm@63612
   422
    let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
wenzelm@63612
   423
    from a' have "?phi a'" by auto
wenzelm@63612
   424
    then have "?phi (SOME b'. ?phi b')"
wenzelm@63612
   425
      using someI[of ?phi] by blast
wenzelm@63612
   426
    then show ?thesis by (auto simp: inv_into_def)
wenzelm@63612
   427
  qed
wenzelm@63612
   428
  ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'"
wenzelm@63612
   429
    by auto
hoelzl@40703
   430
qed
hoelzl@40703
   431
hoelzl@40703
   432
lemma Ex_inj_on_UNION_Sigma:
wenzelm@63629
   433
  "\<exists>f. (inj_on f (\<Union>i \<in> I. A i) \<and> f ` (\<Union>i \<in> I. A i) \<subseteq> (SIGMA i : I. A i))"
hoelzl@40703
   434
proof
wenzelm@63612
   435
  let ?phi = "\<lambda>a i. i \<in> I \<and> a \<in> A i"
wenzelm@63612
   436
  let ?sm = "\<lambda>a. SOME i. ?phi a i"
hoelzl@40703
   437
  let ?f = "\<lambda>a. (?sm a, a)"
wenzelm@63612
   438
  have "inj_on ?f (\<Union>i \<in> I. A i)"
wenzelm@63612
   439
    by (auto simp: inj_on_def)
hoelzl@40703
   440
  moreover
wenzelm@63612
   441
  have "?sm a \<in> I \<and> a \<in> A(?sm a)" if "i \<in> I" and "a \<in> A i" for i a
wenzelm@63612
   442
    using that someI[of "?phi a" i] by auto
wenzelm@63629
   443
  then have "?f ` (\<Union>i \<in> I. A i) \<subseteq> (SIGMA i : I. A i)"
wenzelm@63612
   444
    by auto
wenzelm@63629
   445
  ultimately show "inj_on ?f (\<Union>i \<in> I. A i) \<and> ?f ` (\<Union>i \<in> I. A i) \<subseteq> (SIGMA i : I. A i)"
wenzelm@63612
   446
    by auto
hoelzl@40703
   447
qed
hoelzl@40703
   448
haftmann@56608
   449
lemma inv_unique_comp:
haftmann@56608
   450
  assumes fg: "f \<circ> g = id"
haftmann@56608
   451
    and gf: "g \<circ> f = id"
haftmann@56608
   452
  shows "inv f = g"
haftmann@56608
   453
  using fg gf inv_equality[of g f] by (auto simp add: fun_eq_iff)
haftmann@56608
   454
haftmann@56608
   455
wenzelm@60758
   456
subsection \<open>The Cantor-Bernstein Theorem\<close>
hoelzl@40703
   457
hoelzl@40703
   458
lemma Cantor_Bernstein_aux:
wenzelm@63612
   459
  "\<exists>A' h. A' \<subseteq> A \<and>
wenzelm@63612
   460
    (\<forall>a \<in> A'. a \<notin> g ` (B - f ` A')) \<and>
wenzelm@63612
   461
    (\<forall>a \<in> A'. h a = f a) \<and>
wenzelm@63612
   462
    (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g (h a))"
wenzelm@63612
   463
proof -
wenzelm@63612
   464
  define H where "H A' = A - (g ` (B - (f ` A')))" for A'
wenzelm@63612
   465
  have "mono H" unfolding mono_def H_def by blast
wenzelm@63612
   466
  from lfp_unfold [OF this] obtain A' where "H A' = A'" by blast
wenzelm@63612
   467
  then have "A' = A - (g ` (B - (f ` A')))" by (simp add: H_def)
wenzelm@63612
   468
  then have 1: "A' \<subseteq> A"
wenzelm@63612
   469
    and 2: "\<forall>a \<in> A'.  a \<notin> g ` (B - f ` A')"
wenzelm@63612
   470
    and 3: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
wenzelm@63612
   471
    by blast+
wenzelm@63612
   472
  define h where "h a = (if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" for a
wenzelm@63612
   473
  then have 4: "\<forall>a \<in> A'. h a = f a" by simp
wenzelm@63612
   474
  have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g (h a)"
hoelzl@40703
   475
  proof
wenzelm@63612
   476
    fix a
wenzelm@63612
   477
    let ?phi = "\<lambda>b. b \<in> B - (f ` A') \<and> a = g b"
wenzelm@63612
   478
    assume *: "a \<in> A - A'"
wenzelm@63612
   479
    from * have "h a = (SOME b. ?phi b)" by (auto simp: h_def)
wenzelm@63612
   480
    moreover from 3 * have "\<exists>b. ?phi b" by auto
wenzelm@63612
   481
    ultimately show "?phi (h a)"
wenzelm@63612
   482
      using someI_ex[of ?phi] by auto
hoelzl@40703
   483
  qed
wenzelm@63612
   484
  with 1 2 4 show ?thesis by blast
hoelzl@40703
   485
qed
hoelzl@40703
   486
hoelzl@40703
   487
theorem Cantor_Bernstein:
wenzelm@63612
   488
  assumes inj1: "inj_on f A" and sub1: "f ` A \<subseteq> B"
wenzelm@63612
   489
    and inj2: "inj_on g B" and sub2: "g ` B \<subseteq> A"
hoelzl@40703
   490
  shows "\<exists>h. bij_betw h A B"
hoelzl@40703
   491
proof-
wenzelm@63612
   492
  obtain A' and h where "A' \<subseteq> A"
wenzelm@63612
   493
    and 1: "\<forall>a \<in> A'. a \<notin> g ` (B - f ` A')"
wenzelm@63612
   494
    and 2: "\<forall>a \<in> A'. h a = f a"
wenzelm@63612
   495
    and 3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g (h a)"
wenzelm@63612
   496
    using Cantor_Bernstein_aux [of A g B f] by blast
hoelzl@40703
   497
  have "inj_on h A"
hoelzl@40703
   498
  proof (intro inj_onI)
hoelzl@40703
   499
    fix a1 a2
hoelzl@40703
   500
    assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
hoelzl@40703
   501
    show "a1 = a2"
wenzelm@63612
   502
    proof (cases "a1 \<in> A'")
wenzelm@63612
   503
      case True
hoelzl@40703
   504
      show ?thesis
wenzelm@63612
   505
      proof (cases "a2 \<in> A'")
wenzelm@63612
   506
        case True': True
wenzelm@63612
   507
        with True 2 6 have "f a1 = f a2" by auto
wenzelm@63612
   508
        with inj1 \<open>A' \<subseteq> A\<close> True True' show ?thesis
wenzelm@63612
   509
          unfolding inj_on_def by blast
hoelzl@40703
   510
      next
wenzelm@63612
   511
        case False
wenzelm@63612
   512
        with 2 3 5 6 True have False by force
wenzelm@63612
   513
        then show ?thesis ..
hoelzl@40703
   514
      qed
hoelzl@40703
   515
    next
wenzelm@63612
   516
      case False
hoelzl@40703
   517
      show ?thesis
wenzelm@63612
   518
      proof (cases "a2 \<in> A'")
wenzelm@63612
   519
        case True
wenzelm@63612
   520
        with 2 3 4 6 False have False by auto
wenzelm@63612
   521
        then show ?thesis ..
hoelzl@40703
   522
      next
wenzelm@63612
   523
        case False': False
wenzelm@63612
   524
        with False 3 4 5 have "a1 = g (h a1)" "a2 = g (h a2)" by auto
wenzelm@63612
   525
        with 6 show ?thesis by simp
hoelzl@40703
   526
      qed
hoelzl@40703
   527
    qed
hoelzl@40703
   528
  qed
wenzelm@63612
   529
  moreover have "h ` A = B"
hoelzl@40703
   530
  proof safe
wenzelm@63612
   531
    fix a
wenzelm@63612
   532
    assume "a \<in> A"
wenzelm@63612
   533
    with sub1 2 3 show "h a \<in> B" by (cases "a \<in> A'") auto
hoelzl@40703
   534
  next
wenzelm@63612
   535
    fix b
wenzelm@63612
   536
    assume *: "b \<in> B"
hoelzl@40703
   537
    show "b \<in> h ` A"
wenzelm@63612
   538
    proof (cases "b \<in> f ` A'")
wenzelm@63612
   539
      case True
wenzelm@63612
   540
      then obtain a where "a \<in> A'" "b = f a" by blast
wenzelm@63612
   541
      with \<open>A' \<subseteq> A\<close> 2 show ?thesis by force
hoelzl@40703
   542
    next
wenzelm@63612
   543
      case False
wenzelm@63612
   544
      with 1 * have "g b \<notin> A'" by auto
wenzelm@63612
   545
      with sub2 * have 4: "g b \<in> A - A'" by auto
wenzelm@63612
   546
      with 3 have "h (g b) \<in> B" "g (h (g b)) = g b" by auto
wenzelm@63612
   547
      with inj2 * have "h (g b) = b" by (auto simp: inj_on_def)
wenzelm@63612
   548
      with 4 show ?thesis by force
hoelzl@40703
   549
    qed
hoelzl@40703
   550
  qed
wenzelm@63612
   551
  ultimately show ?thesis
wenzelm@63612
   552
    by (auto simp: bij_betw_def)
hoelzl@40703
   553
qed
paulson@14760
   554
wenzelm@63612
   555
wenzelm@60758
   556
subsection \<open>Other Consequences of Hilbert's Epsilon\<close>
paulson@14760
   557
wenzelm@60758
   558
text \<open>Hilbert's Epsilon and the @{term split} Operator\<close>
paulson@14760
   559
wenzelm@63612
   560
text \<open>Looping simprule!\<close>
wenzelm@63612
   561
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a, b). P (a, b))"
haftmann@26347
   562
  by simp
paulson@14760
   563
haftmann@61424
   564
lemma Eps_case_prod: "Eps (case_prod P) = (SOME xy. P (fst xy) (snd xy))"
haftmann@26347
   565
  by (simp add: split_def)
paulson@14760
   566
wenzelm@63612
   567
lemma Eps_case_prod_eq [simp]: "(SOME (x', y'). x = x' \<and> y = y') = (x, y)"
haftmann@26347
   568
  by blast
paulson@14760
   569
paulson@14760
   570
wenzelm@63612
   571
text \<open>A relation is wellfounded iff it has no infinite descending chain.\<close>
wenzelm@63612
   572
lemma wf_iff_no_infinite_down_chain: "wf r \<longleftrightarrow> (\<not> (\<exists>f. \<forall>i. (f (Suc i), f i) \<in> r))"
wenzelm@63612
   573
  apply (simp only: wf_eq_minimal)
wenzelm@63612
   574
  apply (rule iffI)
wenzelm@63612
   575
   apply (rule notI)
wenzelm@63612
   576
   apply (erule exE)
wenzelm@63612
   577
   apply (erule_tac x = "{w. \<exists>i. w = f i}" in allE)
wenzelm@63612
   578
   apply blast
wenzelm@63612
   579
  apply (erule contrapos_np)
wenzelm@63612
   580
  apply simp
wenzelm@63612
   581
  apply clarify
wenzelm@63612
   582
  apply (subgoal_tac "\<forall>n. rec_nat x (\<lambda>i y. SOME z. z \<in> Q \<and> (z, y) \<in> r) n \<in> Q")
wenzelm@63612
   583
   apply (rule_tac x = "rec_nat x (\<lambda>i y. SOME z. z \<in> Q \<and> (z, y) \<in> r)" in exI)
wenzelm@63612
   584
   apply (rule allI)
wenzelm@63612
   585
   apply simp
wenzelm@63612
   586
   apply (rule someI2_ex)
wenzelm@63612
   587
    apply blast
wenzelm@63612
   588
   apply blast
wenzelm@63612
   589
  apply (rule allI)
wenzelm@63612
   590
  apply (induct_tac n)
wenzelm@63612
   591
   apply simp_all
wenzelm@63612
   592
  apply (rule someI2_ex)
wenzelm@63612
   593
   apply blast
wenzelm@63612
   594
  apply blast
wenzelm@63612
   595
  done
paulson@14760
   596
nipkow@27760
   597
lemma wf_no_infinite_down_chainE:
wenzelm@63612
   598
  assumes "wf r"
wenzelm@63612
   599
  obtains k where "(f (Suc k), f k) \<notin> r"
wenzelm@63612
   600
  using assms wf_iff_no_infinite_down_chain[of r] by blast
nipkow@27760
   601
nipkow@27760
   602
wenzelm@63612
   603
text \<open>A dynamically-scoped fact for TFL\<close>
wenzelm@63612
   604
lemma tfl_some: "\<forall>P x. P x \<longrightarrow> P (Eps P)"
wenzelm@12298
   605
  by (blast intro: someI)
paulson@11451
   606
wenzelm@12298
   607
wenzelm@60758
   608
subsection \<open>Least value operator\<close>
paulson@11451
   609
wenzelm@63612
   610
definition LeastM :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a"
wenzelm@63612
   611
  where "LeastM m P \<equiv> (SOME x. P x \<and> (\<forall>y. P y \<longrightarrow> m x \<le> m y))"
paulson@11451
   612
paulson@11451
   613
syntax
wenzelm@63612
   614
  "_LeastM" :: "pttrn \<Rightarrow> ('a \<Rightarrow> 'b::ord) \<Rightarrow> bool \<Rightarrow> 'a"  ("LEAST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   615
translations
wenzelm@63612
   616
  "LEAST x WRT m. P" \<rightleftharpoons> "CONST LeastM m (\<lambda>x. P)"
paulson@11451
   617
paulson@11451
   618
lemma LeastMI2:
wenzelm@63612
   619
  "P x \<Longrightarrow>
wenzelm@63612
   620
    (\<And>y. P y \<Longrightarrow> m x \<le> m y) \<Longrightarrow>
wenzelm@63612
   621
    (\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> m x \<le> m y \<Longrightarrow> Q x) \<Longrightarrow>
wenzelm@63612
   622
    Q (LeastM m P)"
paulson@14760
   623
  apply (simp add: LeastM_def)
wenzelm@63612
   624
  apply (rule someI2_ex)
wenzelm@63612
   625
   apply blast
wenzelm@63612
   626
  apply blast
wenzelm@12298
   627
  done
paulson@11451
   628
wenzelm@63629
   629
lemma LeastM_equality: "P k \<Longrightarrow> (\<And>x. P x \<Longrightarrow> m k \<le> m x) \<Longrightarrow> m (LEAST x WRT m. P x) = m k"
wenzelm@63629
   630
  for m :: "_ \<Rightarrow> 'a::order"
wenzelm@63612
   631
  apply (rule LeastMI2)
wenzelm@63612
   632
    apply assumption
wenzelm@63612
   633
   apply blast
wenzelm@12298
   634
  apply (blast intro!: order_antisym)
wenzelm@12298
   635
  done
paulson@11451
   636
paulson@11454
   637
lemma wf_linord_ex_has_least:
wenzelm@63612
   638
  "wf r \<Longrightarrow> \<forall>x y. (x, y) \<in> r\<^sup>+ \<longleftrightarrow> (y, x) \<notin> r\<^sup>* \<Longrightarrow> P k \<Longrightarrow> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> (m x, m y) \<in> r\<^sup>*)"
wenzelm@12298
   639
  apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
wenzelm@63612
   640
  apply (drule_tac x = "m ` Collect P" in spec)
wenzelm@63612
   641
  apply force
wenzelm@12298
   642
  done
paulson@11454
   643
wenzelm@63629
   644
lemma ex_has_least_nat: "P k \<Longrightarrow> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> m x \<le> m y)"
wenzelm@63629
   645
  for m :: "'a \<Rightarrow> nat"
wenzelm@12298
   646
  apply (simp only: pred_nat_trancl_eq_le [symmetric])
wenzelm@12298
   647
  apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
wenzelm@63612
   648
   apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le)
wenzelm@63612
   649
  apply assumption
wenzelm@12298
   650
  done
paulson@11454
   651
wenzelm@63629
   652
lemma LeastM_nat_lemma: "P k \<Longrightarrow> P (LeastM m P) \<and> (\<forall>y. P y \<longrightarrow> m (LeastM m P) \<le> m y)"
wenzelm@63629
   653
  for m :: "'a \<Rightarrow> nat"
paulson@14760
   654
  apply (simp add: LeastM_def)
wenzelm@12298
   655
  apply (rule someI_ex)
wenzelm@12298
   656
  apply (erule ex_has_least_nat)
wenzelm@12298
   657
  done
paulson@11454
   658
wenzelm@45607
   659
lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1]
paulson@11454
   660
wenzelm@63629
   661
lemma LeastM_nat_le: "P x \<Longrightarrow> m (LeastM m P) \<le> m x"
wenzelm@63629
   662
  for m :: "'a \<Rightarrow> nat"
wenzelm@63612
   663
  by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp])
paulson@11454
   664
paulson@11451
   665
wenzelm@60758
   666
subsection \<open>Greatest value operator\<close>
paulson@11451
   667
wenzelm@63612
   668
definition GreatestM :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a"
wenzelm@63612
   669
  where "GreatestM m P \<equiv> SOME x. P x \<and> (\<forall>y. P y \<longrightarrow> m y \<le> m x)"
wenzelm@12298
   670
wenzelm@63612
   671
definition Greatest :: "('a::ord \<Rightarrow> bool) \<Rightarrow> 'a"  (binder "GREATEST " 10)
wenzelm@63612
   672
  where "Greatest \<equiv> GreatestM (\<lambda>x. x)"
paulson@11451
   673
paulson@11451
   674
syntax
wenzelm@63612
   675
  "_GreatestM" :: "pttrn \<Rightarrow> ('a \<Rightarrow> 'b::ord) \<Rightarrow> bool \<Rightarrow> 'a"  ("GREATEST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   676
translations
wenzelm@63612
   677
  "GREATEST x WRT m. P" \<rightleftharpoons> "CONST GreatestM m (\<lambda>x. P)"
paulson@11451
   678
paulson@11451
   679
lemma GreatestMI2:
wenzelm@63612
   680
  "P x \<Longrightarrow>
wenzelm@63612
   681
    (\<And>y. P y \<Longrightarrow> m y \<le> m x) \<Longrightarrow>
wenzelm@63612
   682
    (\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> m y \<le> m x \<Longrightarrow> Q x) \<Longrightarrow>
wenzelm@63612
   683
    Q (GreatestM m P)"
paulson@14760
   684
  apply (simp add: GreatestM_def)
wenzelm@63612
   685
  apply (rule someI2_ex)
wenzelm@63612
   686
   apply blast
wenzelm@63612
   687
  apply blast
wenzelm@12298
   688
  done
paulson@11451
   689
wenzelm@63629
   690
lemma GreatestM_equality: "P k \<Longrightarrow> (\<And>x. P x \<Longrightarrow> m x \<le> m k) \<Longrightarrow> m (GREATEST x WRT m. P x) = m k"
wenzelm@63629
   691
  for m :: "_ \<Rightarrow> 'a::order"
wenzelm@63612
   692
  apply (rule GreatestMI2 [where m = m])
wenzelm@63612
   693
    apply assumption
wenzelm@63612
   694
   apply blast
wenzelm@12298
   695
  apply (blast intro!: order_antisym)
wenzelm@12298
   696
  done
paulson@11451
   697
wenzelm@63612
   698
lemma Greatest_equality: "P k \<Longrightarrow> (\<And>x. P x \<Longrightarrow> x \<le> k) \<Longrightarrow> (GREATEST x. P x) = k"
wenzelm@63612
   699
  for k :: "'a::order"
paulson@14760
   700
  apply (simp add: Greatest_def)
wenzelm@63612
   701
  apply (erule GreatestM_equality)
wenzelm@63612
   702
  apply blast
wenzelm@12298
   703
  done
paulson@11451
   704
paulson@11451
   705
lemma ex_has_greatest_nat_lemma:
wenzelm@63629
   706
  "P k \<Longrightarrow> \<forall>x. P x \<longrightarrow> (\<exists>y. P y \<and> \<not> m y \<le> m x) \<Longrightarrow> \<exists>y. P y \<and> \<not> m y < m k + n"
wenzelm@63629
   707
  for m :: "'a \<Rightarrow> nat"
wenzelm@63612
   708
  by (induct n) (force simp: le_Suc_eq)+
paulson@11451
   709
wenzelm@12298
   710
lemma ex_has_greatest_nat:
wenzelm@63629
   711
  "P k \<Longrightarrow> \<forall>y. P y \<longrightarrow> m y < b \<Longrightarrow> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> m y \<le> m x)"
wenzelm@63629
   712
  for m :: "'a \<Rightarrow> nat"
wenzelm@12298
   713
  apply (rule ccontr)
wenzelm@12298
   714
  apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
wenzelm@63612
   715
    apply (subgoal_tac [3] "m k \<le> b")
wenzelm@63612
   716
     apply auto
wenzelm@12298
   717
  done
paulson@11451
   718
wenzelm@12298
   719
lemma GreatestM_nat_lemma:
wenzelm@63629
   720
  "P k \<Longrightarrow> \<forall>y. P y \<longrightarrow> m y < b \<Longrightarrow> P (GreatestM m P) \<and> (\<forall>y. P y \<longrightarrow> m y \<le> m (GreatestM m P))"
wenzelm@63629
   721
  for m :: "'a \<Rightarrow> nat"
paulson@14760
   722
  apply (simp add: GreatestM_def)
wenzelm@12298
   723
  apply (rule someI_ex)
wenzelm@63612
   724
  apply (erule ex_has_greatest_nat)
wenzelm@63612
   725
  apply assumption
wenzelm@12298
   726
  done
paulson@11451
   727
wenzelm@45607
   728
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]
paulson@11451
   729
wenzelm@63629
   730
lemma GreatestM_nat_le: "P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> m y < b \<Longrightarrow> m x \<le> m (GreatestM m P)"
wenzelm@63629
   731
  for m :: "'a \<Rightarrow> nat"
wenzelm@63612
   732
  by (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
wenzelm@12298
   733
wenzelm@12298
   734
wenzelm@63612
   735
text \<open>\<^medskip> Specialization to \<open>GREATEST\<close>.\<close>
wenzelm@12298
   736
wenzelm@63612
   737
lemma GreatestI: "P k \<Longrightarrow> \<forall>y. P y \<longrightarrow> y < b \<Longrightarrow> P (GREATEST x. P x)"
wenzelm@63612
   738
  for k :: nat
wenzelm@63612
   739
  unfolding Greatest_def by (rule GreatestM_natI) auto
paulson@11451
   740
wenzelm@63612
   741
lemma Greatest_le: "P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> y < b \<Longrightarrow> x \<le> (GREATEST x. P x)"
wenzelm@63612
   742
  for x :: nat
wenzelm@63612
   743
  unfolding Greatest_def by (rule GreatestM_nat_le) auto
wenzelm@12298
   744
wenzelm@12298
   745
wenzelm@60758
   746
subsection \<open>An aside: bounded accessible part\<close>
haftmann@49948
   747
wenzelm@60758
   748
text \<open>Finite monotone eventually stable sequences\<close>
haftmann@49948
   749
haftmann@49948
   750
lemma finite_mono_remains_stable_implies_strict_prefix:
haftmann@49948
   751
  fixes f :: "nat \<Rightarrow> 'a::order"
wenzelm@63612
   752
  assumes S: "finite (range f)" "mono f"
wenzelm@63612
   753
    and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
haftmann@49948
   754
  shows "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m < f n) \<and> (\<forall>n\<ge>N. f N = f n)"
haftmann@49948
   755
  using assms
haftmann@49948
   756
proof -
haftmann@49948
   757
  have "\<exists>n. f n = f (Suc n)"
haftmann@49948
   758
  proof (rule ccontr)
haftmann@49948
   759
    assume "\<not> ?thesis"
haftmann@49948
   760
    then have "\<And>n. f n \<noteq> f (Suc n)" by auto
wenzelm@63612
   761
    with \<open>mono f\<close> have "\<And>n. f n < f (Suc n)"
wenzelm@63612
   762
      by (auto simp: le_less mono_iff_le_Suc)
wenzelm@63612
   763
    with lift_Suc_mono_less_iff[of f] have *: "\<And>n m. n < m \<Longrightarrow> f n < f m"
wenzelm@63612
   764
      by auto
traytel@55811
   765
    have "inj f"
traytel@55811
   766
    proof (intro injI)
traytel@55811
   767
      fix x y
traytel@55811
   768
      assume "f x = f y"
wenzelm@63612
   769
      then show "x = y"
wenzelm@63612
   770
        by (cases x y rule: linorder_cases) (auto dest: *)
traytel@55811
   771
    qed
wenzelm@60758
   772
    with \<open>finite (range f)\<close> have "finite (UNIV::nat set)"
haftmann@49948
   773
      by (rule finite_imageD)
haftmann@49948
   774
    then show False by simp
haftmann@49948
   775
  qed
haftmann@49948
   776
  then obtain n where n: "f n = f (Suc n)" ..
wenzelm@63040
   777
  define N where "N = (LEAST n. f n = f (Suc n))"
haftmann@49948
   778
  have N: "f N = f (Suc N)"
haftmann@49948
   779
    unfolding N_def using n by (rule LeastI)
haftmann@49948
   780
  show ?thesis
haftmann@49948
   781
  proof (intro exI[of _ N] conjI allI impI)
wenzelm@63612
   782
    fix n
wenzelm@63612
   783
    assume "N \<le> n"
haftmann@49948
   784
    then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
haftmann@49948
   785
    proof (induct rule: dec_induct)
wenzelm@63612
   786
      case base
wenzelm@63612
   787
      then show ?case by simp
wenzelm@63612
   788
    next
wenzelm@63612
   789
      case (step n)
wenzelm@63612
   790
      then show ?case
wenzelm@63612
   791
        using eq [rule_format, of "n - 1"] N
haftmann@49948
   792
        by (cases n) (auto simp add: le_Suc_eq)
wenzelm@63612
   793
    qed
wenzelm@60758
   794
    from this[of n] \<open>N \<le> n\<close> show "f N = f n" by auto
haftmann@49948
   795
  next
wenzelm@63612
   796
    fix n m :: nat
wenzelm@63612
   797
    assume "m < n" "n \<le> N"
haftmann@49948
   798
    then show "f m < f n"
wenzelm@62683
   799
    proof (induct rule: less_Suc_induct)
haftmann@49948
   800
      case (1 i)
haftmann@49948
   801
      then have "i < N" by simp
haftmann@49948
   802
      then have "f i \<noteq> f (Suc i)"
haftmann@49948
   803
        unfolding N_def by (rule not_less_Least)
wenzelm@60758
   804
      with \<open>mono f\<close> show ?case by (simp add: mono_iff_le_Suc less_le)
wenzelm@63612
   805
    next
wenzelm@63612
   806
      case 2
wenzelm@63612
   807
      then show ?case by simp
wenzelm@63612
   808
    qed
haftmann@49948
   809
  qed
haftmann@49948
   810
qed
haftmann@49948
   811
haftmann@49948
   812
lemma finite_mono_strict_prefix_implies_finite_fixpoint:
haftmann@49948
   813
  fixes f :: "nat \<Rightarrow> 'a set"
haftmann@49948
   814
  assumes S: "\<And>i. f i \<subseteq> S" "finite S"
wenzelm@63612
   815
    and ex: "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n) \<and> (\<forall>n\<ge>N. f N = f n)"
haftmann@49948
   816
  shows "f (card S) = (\<Union>n. f n)"
haftmann@49948
   817
proof -
wenzelm@63612
   818
  from ex obtain N where inj: "\<And>n m. n \<le> N \<Longrightarrow> m \<le> N \<Longrightarrow> m < n \<Longrightarrow> f m \<subset> f n"
wenzelm@63612
   819
    and eq: "\<forall>n\<ge>N. f N = f n"
wenzelm@63612
   820
    by atomize auto
wenzelm@63612
   821
  have "i \<le> N \<Longrightarrow> i \<le> card (f i)" for i
wenzelm@63612
   822
  proof (induct i)
wenzelm@63612
   823
    case 0
wenzelm@63612
   824
    then show ?case by simp
wenzelm@63612
   825
  next
wenzelm@63612
   826
    case (Suc i)
wenzelm@63612
   827
    with inj [of "Suc i" i] have "(f i) \<subset> (f (Suc i))" by auto
wenzelm@63612
   828
    moreover have "finite (f (Suc i))" using S by (rule finite_subset)
wenzelm@63612
   829
    ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
wenzelm@63612
   830
    with Suc inj show ?case by auto
wenzelm@63612
   831
  qed
haftmann@49948
   832
  then have "N \<le> card (f N)" by simp
haftmann@49948
   833
  also have "\<dots> \<le> card S" using S by (intro card_mono)
haftmann@49948
   834
  finally have "f (card S) = f N" using eq by auto
wenzelm@63612
   835
  then show ?thesis
wenzelm@63612
   836
    using eq inj [of N]
haftmann@49948
   837
    apply auto
haftmann@49948
   838
    apply (case_tac "n < N")
wenzelm@63612
   839
     apply (auto simp: not_less)
haftmann@49948
   840
    done
haftmann@49948
   841
qed
haftmann@49948
   842
haftmann@49948
   843
wenzelm@60758
   844
subsection \<open>More on injections, bijections, and inverses\<close>
blanchet@55020
   845
haftmann@63374
   846
locale bijection =
haftmann@63374
   847
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@63374
   848
  assumes bij: "bij f"
haftmann@63374
   849
begin
haftmann@63374
   850
wenzelm@63612
   851
lemma bij_inv: "bij (inv f)"
haftmann@63374
   852
  using bij by (rule bij_imp_bij_inv)
haftmann@63374
   853
wenzelm@63612
   854
lemma surj [simp]: "surj f"
haftmann@63374
   855
  using bij by (rule bij_is_surj)
haftmann@63374
   856
wenzelm@63612
   857
lemma inj: "inj f"
haftmann@63374
   858
  using bij by (rule bij_is_inj)
haftmann@63374
   859
wenzelm@63612
   860
lemma surj_inv [simp]: "surj (inv f)"
haftmann@63374
   861
  using inj by (rule inj_imp_surj_inv)
haftmann@63374
   862
wenzelm@63612
   863
lemma inj_inv: "inj (inv f)"
haftmann@63374
   864
  using surj by (rule surj_imp_inj_inv)
haftmann@63374
   865
wenzelm@63612
   866
lemma eqI: "f a = f b \<Longrightarrow> a = b"
haftmann@63374
   867
  using inj by (rule injD)
haftmann@63374
   868
wenzelm@63612
   869
lemma eq_iff [simp]: "f a = f b \<longleftrightarrow> a = b"
haftmann@63374
   870
  by (auto intro: eqI)
haftmann@63374
   871
wenzelm@63612
   872
lemma eq_invI: "inv f a = inv f b \<Longrightarrow> a = b"
haftmann@63374
   873
  using inj_inv by (rule injD)
haftmann@63374
   874
wenzelm@63612
   875
lemma eq_inv_iff [simp]: "inv f a = inv f b \<longleftrightarrow> a = b"
haftmann@63374
   876
  by (auto intro: eq_invI)
haftmann@63374
   877
wenzelm@63612
   878
lemma inv_left [simp]: "inv f (f a) = a"
haftmann@63374
   879
  using inj by (simp add: inv_f_eq)
haftmann@63374
   880
wenzelm@63612
   881
lemma inv_comp_left [simp]: "inv f \<circ> f = id"
haftmann@63374
   882
  by (simp add: fun_eq_iff)
haftmann@63374
   883
wenzelm@63612
   884
lemma inv_right [simp]: "f (inv f a) = a"
haftmann@63374
   885
  using surj by (simp add: surj_f_inv_f)
haftmann@63374
   886
wenzelm@63612
   887
lemma inv_comp_right [simp]: "f \<circ> inv f = id"
haftmann@63374
   888
  by (simp add: fun_eq_iff)
haftmann@63374
   889
wenzelm@63612
   890
lemma inv_left_eq_iff [simp]: "inv f a = b \<longleftrightarrow> f b = a"
haftmann@63374
   891
  by auto
haftmann@63374
   892
wenzelm@63612
   893
lemma inv_right_eq_iff [simp]: "b = inv f a \<longleftrightarrow> f b = a"
haftmann@63374
   894
  by auto
haftmann@63374
   895
haftmann@63374
   896
end
haftmann@63374
   897
blanchet@55020
   898
lemma infinite_imp_bij_betw:
wenzelm@63612
   899
  assumes infinite: "\<not> finite A"
wenzelm@63612
   900
  shows "\<exists>h. bij_betw h A (A - {a})"
wenzelm@63612
   901
proof (cases "a \<in> A")
wenzelm@63612
   902
  case False
wenzelm@63612
   903
  then have "A - {a} = A" by blast
wenzelm@63612
   904
  then show ?thesis
wenzelm@63612
   905
    using bij_betw_id[of A] by auto
blanchet@55020
   906
next
wenzelm@63612
   907
  case True
wenzelm@63612
   908
  with infinite have "\<not> finite (A - {a})" by auto
wenzelm@63612
   909
  with infinite_iff_countable_subset[of "A - {a}"]
wenzelm@63612
   910
  obtain f :: "nat \<Rightarrow> 'a" where 1: "inj f" and 2: "f ` UNIV \<subseteq> A - {a}" by blast
wenzelm@63612
   911
  define g where "g n = (if n = 0 then a else f (Suc n))" for n
wenzelm@63612
   912
  define A' where "A' = g ` UNIV"
wenzelm@63612
   913
  have *: "\<forall>y. f y \<noteq> a" using 2 by blast
wenzelm@63612
   914
  have 3: "inj_on g UNIV \<and> g ` UNIV \<subseteq> A \<and> a \<in> g ` UNIV"
wenzelm@63612
   915
    apply (auto simp add: True g_def [abs_def])
wenzelm@63612
   916
     apply (unfold inj_on_def)
wenzelm@63612
   917
     apply (intro ballI impI)
wenzelm@63612
   918
     apply (case_tac "x = 0")
wenzelm@63612
   919
      apply (auto simp add: 2)
wenzelm@63612
   920
  proof -
wenzelm@63612
   921
    fix y
wenzelm@63612
   922
    assume "a = (if y = 0 then a else f (Suc y))"
wenzelm@63612
   923
    then show "y = 0" by (cases "y = 0") (use * in auto)
blanchet@55020
   924
  next
blanchet@55020
   925
    fix x y
blanchet@55020
   926
    assume "f (Suc x) = (if y = 0 then a else f (Suc y))"
wenzelm@63612
   927
    with 1 * show "x = y" by (cases "y = 0") (auto simp: inj_on_def)
blanchet@55020
   928
  next
wenzelm@63612
   929
    fix n
wenzelm@63612
   930
    from 2 show "f (Suc n) \<in> A" by blast
blanchet@55020
   931
  qed
wenzelm@63612
   932
  then have 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<subseteq> A"
wenzelm@63612
   933
    using inj_on_imp_bij_betw[of g] by (auto simp: A'_def)
wenzelm@63612
   934
  then have 5: "bij_betw (inv g) A' UNIV"
wenzelm@63612
   935
    by (auto simp add: bij_betw_inv_into)
wenzelm@63612
   936
  from 3 obtain n where n: "g n = a" by auto
wenzelm@63612
   937
  have 6: "bij_betw g (UNIV - {n}) (A' - {a})"
wenzelm@63612
   938
    by (rule bij_betw_subset) (use 3 4 n in \<open>auto simp: image_set_diff A'_def\<close>)
wenzelm@63612
   939
  define v where "v m = (if m < n then m else Suc m)" for m
blanchet@55020
   940
  have 7: "bij_betw v UNIV (UNIV - {n})"
wenzelm@63612
   941
  proof (unfold bij_betw_def inj_on_def, intro conjI, clarify)
wenzelm@63612
   942
    fix m1 m2
wenzelm@63612
   943
    assume "v m1 = v m2"
wenzelm@63612
   944
    then show "m1 = m2"
wenzelm@63612
   945
      apply (cases "m1 < n")
wenzelm@63612
   946
       apply (cases "m2 < n")
wenzelm@63612
   947
        apply (auto simp: inj_on_def v_def [abs_def])
wenzelm@63612
   948
      apply (cases "m2 < n")
wenzelm@63612
   949
       apply auto
wenzelm@63612
   950
      done
blanchet@55020
   951
  next
blanchet@55020
   952
    show "v ` UNIV = UNIV - {n}"
wenzelm@63612
   953
    proof (auto simp: v_def [abs_def])
wenzelm@63612
   954
      fix m
wenzelm@63612
   955
      assume "m \<noteq> n"
wenzelm@63612
   956
      assume *: "m \<notin> Suc ` {m'. \<not> m' < n}"
wenzelm@63612
   957
      have False if "n \<le> m"
wenzelm@63612
   958
      proof -
wenzelm@63612
   959
        from \<open>m \<noteq> n\<close> that have **: "Suc n \<le> m" by auto
wenzelm@63612
   960
        from Suc_le_D [OF this] obtain m' where m': "m = Suc m'" ..
wenzelm@63612
   961
        with ** have "n \<le> m'" by auto
wenzelm@63612
   962
        with m' * show ?thesis by auto
wenzelm@63612
   963
      qed
wenzelm@63612
   964
      then show "m < n" by force
blanchet@55020
   965
    qed
blanchet@55020
   966
  qed
wenzelm@63612
   967
  define h' where "h' = g \<circ> v \<circ> (inv g)"
wenzelm@63612
   968
  with 5 6 7 have 8: "bij_betw h' A' (A' - {a})"
wenzelm@63612
   969
    by (auto simp add: bij_betw_trans)
wenzelm@63612
   970
  define h where "h b = (if b \<in> A' then h' b else b)" for b
wenzelm@63612
   971
  then have "\<forall>b \<in> A'. h b = h' b" by simp
wenzelm@63612
   972
  with 8 have "bij_betw h  A' (A' - {a})"
wenzelm@63612
   973
    using bij_betw_cong[of A' h] by auto
blanchet@55020
   974
  moreover
wenzelm@63612
   975
  have "\<forall>b \<in> A - A'. h b = b" by (auto simp: h_def)
wenzelm@63612
   976
  then have "bij_betw h  (A - A') (A - A')"
wenzelm@63612
   977
    using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto
blanchet@55020
   978
  moreover
wenzelm@63612
   979
  from 4 have "(A' \<inter> (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>
wenzelm@63612
   980
    ((A' - {a}) \<inter> (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"
wenzelm@63612
   981
    by blast
blanchet@55020
   982
  ultimately have "bij_betw h A (A - {a})"
wenzelm@63612
   983
    using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp
wenzelm@63612
   984
  then show ?thesis by blast
blanchet@55020
   985
qed
blanchet@55020
   986
blanchet@55020
   987
lemma infinite_imp_bij_betw2:
wenzelm@63612
   988
  assumes "\<not> finite A"
wenzelm@63612
   989
  shows "\<exists>h. bij_betw h A (A \<union> {a})"
wenzelm@63612
   990
proof (cases "a \<in> A")
wenzelm@63612
   991
  case True
wenzelm@63612
   992
  then have "A \<union> {a} = A" by blast
wenzelm@63612
   993
  then show ?thesis using bij_betw_id[of A] by auto
blanchet@55020
   994
next
wenzelm@63612
   995
  case False
blanchet@55020
   996
  let ?A' = "A \<union> {a}"
wenzelm@63612
   997
  from False have "A = ?A' - {a}" by blast
wenzelm@63612
   998
  moreover from assms have "\<not> finite ?A'" by auto
blanchet@55020
   999
  ultimately obtain f where "bij_betw f ?A' A"
wenzelm@63612
  1000
    using infinite_imp_bij_betw[of ?A' a] by auto
wenzelm@63612
  1001
  then have "bij_betw (inv_into ?A' f) A ?A'" by (rule bij_betw_inv_into)
wenzelm@63612
  1002
  then show ?thesis by auto
blanchet@55020
  1003
qed
blanchet@55020
  1004
wenzelm@63612
  1005
lemma bij_betw_inv_into_left: "bij_betw f A A' \<Longrightarrow> a \<in> A \<Longrightarrow> inv_into A f (f a) = a"
wenzelm@63612
  1006
  unfolding bij_betw_def by clarify (rule inv_into_f_f)
blanchet@55020
  1007
wenzelm@63612
  1008
lemma bij_betw_inv_into_right: "bij_betw f A A' \<Longrightarrow> a' \<in> A' \<Longrightarrow> f (inv_into A f a') = a'"
wenzelm@63612
  1009
  unfolding bij_betw_def using f_inv_into_f by force
blanchet@55020
  1010
blanchet@55020
  1011
lemma bij_betw_inv_into_subset:
wenzelm@63612
  1012
  "bij_betw f A A' \<Longrightarrow> B \<subseteq> A \<Longrightarrow> f ` B = B' \<Longrightarrow> bij_betw (inv_into A f) B' B"
wenzelm@63612
  1013
  by (auto simp: bij_betw_def intro: inj_on_inv_into)
blanchet@55020
  1014
blanchet@55020
  1015
wenzelm@60758
  1016
subsection \<open>Specification package -- Hilbertized version\<close>
wenzelm@17893
  1017
wenzelm@63612
  1018
lemma exE_some: "Ex P \<Longrightarrow> c \<equiv> Eps P \<Longrightarrow> P c"
wenzelm@17893
  1019
  by (simp only: someI_ex)
wenzelm@17893
  1020
wenzelm@48891
  1021
ML_file "Tools/choice_specification.ML"
skalberg@14115
  1022
paulson@11451
  1023
end