src/HOL/Hilbert_Choice.thy
author paulson <lp15@cam.ac.uk>
Mon Feb 22 14:37:56 2016 +0000 (2016-02-22)
changeset 62379 340738057c8c
parent 62343 24106dc44def
child 62521 6383440f41a8
permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
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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 => bool) => 'a" where
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  someI: "P x ==> P (Eps P)"
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syntax (epsilon)
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  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
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syntax (HOL)
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  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
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syntax
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  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
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translations
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  "SOME x. P" == "CONST Eps (%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 => ('a => 'b) => ('b => 'a)" where
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"inv_into A f == %x. SOME y. y : A & f y = x"
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abbreviation inv :: "('a => 'b) => ('b => 'a)" where
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"inv == inv_into UNIV"
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subsection \<open>Hilbert's Epsilon-operator\<close>
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text\<open>Easier to apply than \<open>someI\<close> if the witness comes from an
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existential formula\<close>
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lemma someI_ex [elim?]: "\<exists>x. P x ==> 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>Easier to apply than \<open>someI\<close> because the conclusion has only one
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occurrence of @{term P}.\<close>
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lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
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  by (blast intro: someI)
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text\<open>Easier to apply than \<open>someI2\<close> if the witness comes from an
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existential formula\<close>
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lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> 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; !!x. x \<in> A \<and> P x ==> Q x |] ==> 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]:
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     "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
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by (blast intro: someI2)
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lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
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by blast
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lemma some_eq_ex: "P (SOME x. P x) =  (\<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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apply (rule some_equality)
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apply (rule refl, assumption)
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done
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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 ==> \<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 ==> \<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" and
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           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 def f \<equiv> "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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  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 add: f_def)
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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 = (%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 : f ` A ==> inv_into A f x : A"
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apply (simp add: inv_into_def)
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apply (fast intro: someI2)
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done
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lemma inv_id [simp]: "inv id = id"
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by (simp add: inv_into_def id_def)
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lemma inv_into_f_f [simp]:
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  "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
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apply (simp add: inv_into_def inj_on_def)
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apply (blast intro: someI2)
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done
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lemma inv_f_f: "inj f ==> inv f (f x) = x"
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by simp
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lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
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apply (simp add: inv_into_def)
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apply (fast intro: someI2)
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done
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lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
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apply (erule subst)
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apply (fast intro: inv_into_f_f)
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done
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lemma inv_f_eq: "[| inj f; f x = y |] ==> 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; ALL x. f(g x) = x |] ==> 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 injf: "inj f" and minor: "!!y. y \<in> range(f) ==> 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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  hence "P(inv f (f x))" by (rule minor)
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  thus "P x" by (simp add: inv_into_f_f [OF injf])
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qed
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lemma inj_iff: "(inj f) = (inv f o f = id)"
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apply (simp add: o_def fun_eq_iff)
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apply (blast intro: inj_on_inverseI inv_into_f_f)
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done
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lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
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by (simp add: inj_iff)
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lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
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by (simp add: comp_assoc)
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lemma inv_into_image_cancel[simp]:
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  "inj_on f A ==> S <= A ==> 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 ==> 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 ==> 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: f`A"
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      and y: "y: f`A"
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  shows "x=y"
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proof -
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  have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
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  thus ?thesis by (simp add: f_inv_into_f x y)
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qed
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lemma inj_on_inv_into: "B <= f`A ==> 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 ==> 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 ==> inj (inv f)"
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by (simp add: inj_on_inv_into)
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lemma surj_iff: "(surj f) = (f o 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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  unfolding surj_iff by (simp add: o_def fun_eq_iff)
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lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> 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 ==> 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: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
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apply (rule ext)
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apply (auto simp add: inv_into_def)
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done
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lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
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apply (rule inv_equality)
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apply (auto simp add: bij_def surj_f_inv_f)
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done
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(** bij(inv f) implies little about f.  Consider f::bool=>bool such that
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    f(True)=f(False)=True.  Then it's consistent with axiom someI that
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    inv f could be any function at all, including the identity function.
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    If inv f=id then inv f is a bijection, but inj f, surj(f) and
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    inv(inv f)=f all fail.
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**)
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lemma inv_into_comp:
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  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
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  inv_into A (f o g) x = (inv_into A g o 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; bij g |] ==> inv (f o g) = inv g o inv f"
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apply (rule inv_equality)
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apply (auto simp add: bij_def surj_f_inv_f)
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done
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lemma image_surj_f_inv_f: "surj f ==> 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 ==> inv f ` (f ` A) = A"
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  by simp
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lemma inv_image_comp: "inj f ==> 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 ==> 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 ==> 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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  from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
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  with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
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    by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
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  then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
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  then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
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  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
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  moreover 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" by (simp add: inv_into_def)
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    thus "x \<in> range (\<lambda>f::'a \<Rightarrow> 'b. inv f b1)" by blast
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  qed
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  ultimately show "finite (UNIV :: 'a set)" by simp
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qed
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text \<open>
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  Every infinite set contains a countable subset. More precisely we
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  show that a set \<open>S\<close> is infinite if and only if there exists an
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  injective function from the naturals into \<open>S\<close>.
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  The ``only if'' direction is harder because it requires the
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  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:
traytel@54578
   308
  assumes inf: "\<not> finite (S::'a set)"
traytel@54578
   309
  shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S"
wenzelm@61799
   310
  \<comment> \<open>Courtesy of Stephan Merz\<close>
traytel@54578
   311
proof -
blanchet@55415
   312
  def Sseq \<equiv> "rec_nat S (\<lambda>n T. T - {SOME e. e \<in> T})"
traytel@54578
   313
  def pick \<equiv> "\<lambda>n. (SOME e. e \<in> Sseq n)"
traytel@54578
   314
  { fix n have "Sseq n \<subseteq> S" "\<not> finite (Sseq n)" by (induct n) (auto simp add: Sseq_def inf) }
traytel@55811
   315
  moreover then have *: "\<And>n. pick n \<in> Sseq n"
traytel@55811
   316
    unfolding pick_def by (subst (asm) finite.simps) (auto simp add: ex_in_conv intro: someI_ex)
traytel@54578
   317
  ultimately have "range pick \<subseteq> S" by auto
traytel@54578
   318
  moreover
lp15@60974
   319
  { fix n m
traytel@54578
   320
    have "pick n \<notin> Sseq (n + Suc m)" by (induct m) (auto simp add: Sseq_def pick_def)
traytel@55811
   321
    with * have "pick n \<noteq> pick (n + Suc m)" by auto
traytel@54578
   322
  }
traytel@54578
   323
  then have "inj pick" by (intro linorder_injI) (auto simp add: less_iff_Suc_add)
traytel@54578
   324
  ultimately show ?thesis by blast
traytel@54578
   325
qed
traytel@54578
   326
traytel@54578
   327
lemma infinite_iff_countable_subset: "\<not> finite S \<longleftrightarrow> (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
wenzelm@61799
   328
  \<comment> \<open>Courtesy of Stephan Merz\<close>
traytel@55811
   329
  using finite_imageD finite_subset infinite_UNIV_char_0 infinite_countable_subset by auto
traytel@54578
   330
hoelzl@40703
   331
lemma image_inv_into_cancel:
hoelzl@40703
   332
  assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
hoelzl@40703
   333
  shows "f `((inv_into A f)`B') = B'"
hoelzl@40703
   334
  using assms
hoelzl@40703
   335
proof (auto simp add: f_inv_into_f)
hoelzl@40703
   336
  let ?f' = "(inv_into A f)"
hoelzl@40703
   337
  fix a' assume *: "a' \<in> B'"
hoelzl@40703
   338
  then have "a' \<in> A'" using SUB by auto
hoelzl@40703
   339
  then have "a' = f (?f' a')"
hoelzl@40703
   340
    using SURJ by (auto simp add: f_inv_into_f)
hoelzl@40703
   341
  then show "a' \<in> f ` (?f' ` B')" using * by blast
hoelzl@40703
   342
qed
hoelzl@40703
   343
hoelzl@40703
   344
lemma inv_into_inv_into_eq:
hoelzl@40703
   345
  assumes "bij_betw f A A'" "a \<in> A"
hoelzl@40703
   346
  shows "inv_into A' (inv_into A f) a = f a"
hoelzl@40703
   347
proof -
hoelzl@40703
   348
  let ?f' = "inv_into A f"   let ?f'' = "inv_into A' ?f'"
hoelzl@40703
   349
  have 1: "bij_betw ?f' A' A" using assms
hoelzl@40703
   350
  by (auto simp add: bij_betw_inv_into)
hoelzl@40703
   351
  obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
wenzelm@60758
   352
    using 1 \<open>a \<in> A\<close> unfolding bij_betw_def by force
hoelzl@40703
   353
  hence "?f'' a = a'"
wenzelm@60758
   354
    using \<open>a \<in> A\<close> 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
hoelzl@40703
   355
  moreover have "f a = a'" using assms 2 3
huffman@44921
   356
    by (auto simp add: bij_betw_def)
hoelzl@40703
   357
  ultimately show "?f'' a = f a" by simp
hoelzl@40703
   358
qed
hoelzl@40703
   359
hoelzl@40703
   360
lemma inj_on_iff_surj:
hoelzl@40703
   361
  assumes "A \<noteq> {}"
hoelzl@40703
   362
  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
hoelzl@40703
   363
proof safe
hoelzl@40703
   364
  fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
hoelzl@40703
   365
  let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"  let ?csi = "\<lambda>a. a \<in> A"
hoelzl@40703
   366
  let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
hoelzl@40703
   367
  have "?g ` A' = A"
hoelzl@40703
   368
  proof
hoelzl@40703
   369
    show "?g ` A' \<le> A"
hoelzl@40703
   370
    proof clarify
hoelzl@40703
   371
      fix a' assume *: "a' \<in> A'"
hoelzl@40703
   372
      show "?g a' \<in> A"
hoelzl@40703
   373
      proof cases
hoelzl@40703
   374
        assume Case1: "a' \<in> f ` A"
hoelzl@40703
   375
        then obtain a where "?phi a' a" by blast
hoelzl@40703
   376
        hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
hoelzl@40703
   377
        with Case1 show ?thesis by auto
hoelzl@40703
   378
      next
hoelzl@40703
   379
        assume Case2: "a' \<notin> f ` A"
hoelzl@40703
   380
        hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
hoelzl@40703
   381
        with Case2 show ?thesis by auto
hoelzl@40703
   382
      qed
hoelzl@40703
   383
    qed
hoelzl@40703
   384
  next
hoelzl@40703
   385
    show "A \<le> ?g ` A'"
hoelzl@40703
   386
    proof-
hoelzl@40703
   387
      {fix a assume *: "a \<in> A"
hoelzl@40703
   388
       let ?b = "SOME aa. ?phi (f a) aa"
hoelzl@40703
   389
       have "?phi (f a) a" using * by auto
hoelzl@40703
   390
       hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
hoelzl@40703
   391
       hence "?g(f a) = ?b" using * by auto
hoelzl@40703
   392
       moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
hoelzl@40703
   393
       ultimately have "?g(f a) = a" by simp
hoelzl@40703
   394
       with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
hoelzl@40703
   395
      }
hoelzl@40703
   396
      thus ?thesis by force
hoelzl@40703
   397
    qed
hoelzl@40703
   398
  qed
hoelzl@40703
   399
  thus "\<exists>g. g ` A' = A" by blast
hoelzl@40703
   400
next
hoelzl@40703
   401
  fix g  let ?f = "inv_into A' g"
hoelzl@40703
   402
  have "inj_on ?f (g ` A')"
hoelzl@40703
   403
    by (auto simp add: inj_on_inv_into)
hoelzl@40703
   404
  moreover
hoelzl@40703
   405
  {fix a' assume *: "a' \<in> A'"
hoelzl@40703
   406
   let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
hoelzl@40703
   407
   have "?phi a'" using * by auto
hoelzl@40703
   408
   hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
hoelzl@40703
   409
   hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
hoelzl@40703
   410
  }
hoelzl@40703
   411
  ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
hoelzl@40703
   412
qed
hoelzl@40703
   413
hoelzl@40703
   414
lemma Ex_inj_on_UNION_Sigma:
wenzelm@60585
   415
  "\<exists>f. (inj_on f (\<Union>i \<in> I. A i) \<and> f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i))"
hoelzl@40703
   416
proof
hoelzl@40703
   417
  let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
hoelzl@40703
   418
  let ?sm = "\<lambda> a. SOME i. ?phi a i"
hoelzl@40703
   419
  let ?f = "\<lambda>a. (?sm a, a)"
wenzelm@60585
   420
  have "inj_on ?f (\<Union>i \<in> I. A i)" unfolding inj_on_def by auto
hoelzl@40703
   421
  moreover
hoelzl@40703
   422
  { { fix i a assume "i \<in> I" and "a \<in> A i"
hoelzl@40703
   423
      hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
hoelzl@40703
   424
    }
wenzelm@60585
   425
    hence "?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
hoelzl@40703
   426
  }
hoelzl@40703
   427
  ultimately
wenzelm@60585
   428
  show "inj_on ?f (\<Union>i \<in> I. A i) \<and> ?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)"
hoelzl@40703
   429
  by auto
hoelzl@40703
   430
qed
hoelzl@40703
   431
haftmann@56608
   432
lemma inv_unique_comp:
haftmann@56608
   433
  assumes fg: "f \<circ> g = id"
haftmann@56608
   434
    and gf: "g \<circ> f = id"
haftmann@56608
   435
  shows "inv f = g"
haftmann@56608
   436
  using fg gf inv_equality[of g f] by (auto simp add: fun_eq_iff)
haftmann@56608
   437
haftmann@56608
   438
wenzelm@60758
   439
subsection \<open>The Cantor-Bernstein Theorem\<close>
hoelzl@40703
   440
hoelzl@40703
   441
lemma Cantor_Bernstein_aux:
hoelzl@40703
   442
  shows "\<exists>A' h. A' \<le> A \<and>
hoelzl@40703
   443
                (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
hoelzl@40703
   444
                (\<forall>a \<in> A'. h a = f a) \<and>
hoelzl@40703
   445
                (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
hoelzl@40703
   446
proof-
hoelzl@40703
   447
  obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
hoelzl@40703
   448
  have 0: "mono H" unfolding mono_def H_def by blast
hoelzl@40703
   449
  then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
hoelzl@40703
   450
  hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
hoelzl@40703
   451
  hence 3: "A' \<le> A" by blast
hoelzl@40703
   452
  have 4: "\<forall>a \<in> A'.  a \<notin> g`(B - f ` A')"
hoelzl@40703
   453
  using 2 by blast
hoelzl@40703
   454
  have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
hoelzl@40703
   455
  using 2 by blast
hoelzl@40703
   456
  (*  *)
hoelzl@40703
   457
  obtain h where h_def:
hoelzl@40703
   458
  "h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
hoelzl@40703
   459
  hence "\<forall>a \<in> A'. h a = f a" by auto
hoelzl@40703
   460
  moreover
hoelzl@40703
   461
  have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
hoelzl@40703
   462
  proof
hoelzl@40703
   463
    fix a assume *: "a \<in> A - A'"
hoelzl@40703
   464
    let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
hoelzl@40703
   465
    have "h a = (SOME b. ?phi b)" using h_def * by auto
hoelzl@40703
   466
    moreover have "\<exists>b. ?phi b" using 5 *  by auto
hoelzl@40703
   467
    ultimately show  "?phi (h a)" using someI_ex[of ?phi] by auto
hoelzl@40703
   468
  qed
hoelzl@40703
   469
  ultimately show ?thesis using 3 4 by blast
hoelzl@40703
   470
qed
hoelzl@40703
   471
hoelzl@40703
   472
theorem Cantor_Bernstein:
hoelzl@40703
   473
  assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
hoelzl@40703
   474
          INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
hoelzl@40703
   475
  shows "\<exists>h. bij_betw h A B"
hoelzl@40703
   476
proof-
hoelzl@40703
   477
  obtain A' and h where 0: "A' \<le> A" and
hoelzl@40703
   478
  1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
hoelzl@40703
   479
  2: "\<forall>a \<in> A'. h a = f a" and
hoelzl@40703
   480
  3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
hoelzl@40703
   481
  using Cantor_Bernstein_aux[of A g B f] by blast
hoelzl@40703
   482
  have "inj_on h A"
hoelzl@40703
   483
  proof (intro inj_onI)
hoelzl@40703
   484
    fix a1 a2
hoelzl@40703
   485
    assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
hoelzl@40703
   486
    show "a1 = a2"
hoelzl@40703
   487
    proof(cases "a1 \<in> A'")
hoelzl@40703
   488
      assume Case1: "a1 \<in> A'"
hoelzl@40703
   489
      show ?thesis
hoelzl@40703
   490
      proof(cases "a2 \<in> A'")
hoelzl@40703
   491
        assume Case11: "a2 \<in> A'"
hoelzl@40703
   492
        hence "f a1 = f a2" using Case1 2 6 by auto
hoelzl@40703
   493
        thus ?thesis using INJ1 Case1 Case11 0
hoelzl@40703
   494
        unfolding inj_on_def by blast
hoelzl@40703
   495
      next
hoelzl@40703
   496
        assume Case12: "a2 \<notin> A'"
hoelzl@40703
   497
        hence False using 3 5 2 6 Case1 by force
hoelzl@40703
   498
        thus ?thesis by simp
hoelzl@40703
   499
      qed
hoelzl@40703
   500
    next
hoelzl@40703
   501
    assume Case2: "a1 \<notin> A'"
hoelzl@40703
   502
      show ?thesis
hoelzl@40703
   503
      proof(cases "a2 \<in> A'")
hoelzl@40703
   504
        assume Case21: "a2 \<in> A'"
hoelzl@40703
   505
        hence False using 3 4 2 6 Case2 by auto
hoelzl@40703
   506
        thus ?thesis by simp
hoelzl@40703
   507
      next
hoelzl@40703
   508
        assume Case22: "a2 \<notin> A'"
hoelzl@40703
   509
        hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
hoelzl@40703
   510
        thus ?thesis using 6 by simp
hoelzl@40703
   511
      qed
hoelzl@40703
   512
    qed
hoelzl@40703
   513
  qed
hoelzl@40703
   514
  (*  *)
hoelzl@40703
   515
  moreover
hoelzl@40703
   516
  have "h ` A = B"
hoelzl@40703
   517
  proof safe
hoelzl@40703
   518
    fix a assume "a \<in> A"
wenzelm@47988
   519
    thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto
hoelzl@40703
   520
  next
hoelzl@40703
   521
    fix b assume *: "b \<in> B"
hoelzl@40703
   522
    show "b \<in> h ` A"
hoelzl@40703
   523
    proof(cases "b \<in> f ` A'")
hoelzl@40703
   524
      assume Case1: "b \<in> f ` A'"
hoelzl@40703
   525
      then obtain a where "a \<in> A' \<and> b = f a" by blast
hoelzl@40703
   526
      thus ?thesis using 2 0 by force
hoelzl@40703
   527
    next
hoelzl@40703
   528
      assume Case2: "b \<notin> f ` A'"
hoelzl@40703
   529
      hence "g b \<notin> A'" using 1 * by auto
hoelzl@40703
   530
      hence 4: "g b \<in> A - A'" using * SUB2 by auto
hoelzl@40703
   531
      hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
hoelzl@40703
   532
      using 3 by auto
hoelzl@40703
   533
      hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
hoelzl@40703
   534
      thus ?thesis using 4 by force
hoelzl@40703
   535
    qed
hoelzl@40703
   536
  qed
hoelzl@40703
   537
  (*  *)
hoelzl@40703
   538
  ultimately show ?thesis unfolding bij_betw_def by auto
hoelzl@40703
   539
qed
paulson@14760
   540
wenzelm@60758
   541
subsection \<open>Other Consequences of Hilbert's Epsilon\<close>
paulson@14760
   542
wenzelm@60758
   543
text \<open>Hilbert's Epsilon and the @{term split} Operator\<close>
paulson@14760
   544
wenzelm@60758
   545
text\<open>Looping simprule\<close>
paulson@14760
   546
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
haftmann@26347
   547
  by simp
paulson@14760
   548
haftmann@61424
   549
lemma Eps_case_prod: "Eps (case_prod P) = (SOME xy. P (fst xy) (snd xy))"
haftmann@26347
   550
  by (simp add: split_def)
paulson@14760
   551
haftmann@61424
   552
lemma Eps_case_prod_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
haftmann@26347
   553
  by blast
paulson@14760
   554
paulson@14760
   555
wenzelm@60758
   556
text\<open>A relation is wellfounded iff it has no infinite descending chain\<close>
paulson@14760
   557
lemma wf_iff_no_infinite_down_chain:
paulson@14760
   558
  "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
paulson@14760
   559
apply (simp only: wf_eq_minimal)
paulson@14760
   560
apply (rule iffI)
paulson@14760
   561
 apply (rule notI)
paulson@14760
   562
 apply (erule exE)
paulson@14760
   563
 apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
paulson@14760
   564
apply (erule contrapos_np, simp, clarify)
blanchet@55415
   565
apply (subgoal_tac "\<forall>n. rec_nat x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
blanchet@55415
   566
 apply (rule_tac x = "rec_nat x (%i y. @z. z:Q & (z,y) :r)" in exI)
paulson@14760
   567
 apply (rule allI, simp)
paulson@14760
   568
 apply (rule someI2_ex, blast, blast)
paulson@14760
   569
apply (rule allI)
paulson@14760
   570
apply (induct_tac "n", simp_all)
paulson@14760
   571
apply (rule someI2_ex, blast+)
paulson@14760
   572
done
paulson@14760
   573
nipkow@27760
   574
lemma wf_no_infinite_down_chainE:
nipkow@27760
   575
  assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
wenzelm@60758
   576
using \<open>wf r\<close> wf_iff_no_infinite_down_chain[of r] by blast
nipkow@27760
   577
nipkow@27760
   578
wenzelm@60758
   579
text\<open>A dynamically-scoped fact for TFL\<close>
wenzelm@12298
   580
lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
wenzelm@12298
   581
  by (blast intro: someI)
paulson@11451
   582
wenzelm@12298
   583
wenzelm@60758
   584
subsection \<open>Least value operator\<close>
paulson@11451
   585
haftmann@35416
   586
definition
haftmann@35416
   587
  LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
paulson@14760
   588
  "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
paulson@11451
   589
paulson@11451
   590
syntax
wenzelm@12298
   591
  "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   592
translations
wenzelm@35115
   593
  "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
paulson@11451
   594
paulson@11451
   595
lemma LeastMI2:
wenzelm@12298
   596
  "P x ==> (!!y. P y ==> m x <= m y)
wenzelm@12298
   597
    ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
wenzelm@12298
   598
    ==> Q (LeastM m P)"
paulson@14760
   599
  apply (simp add: LeastM_def)
paulson@14208
   600
  apply (rule someI2_ex, blast, blast)
wenzelm@12298
   601
  done
paulson@11451
   602
paulson@11451
   603
lemma LeastM_equality:
wenzelm@12298
   604
  "P k ==> (!!x. P x ==> m k <= m x)
wenzelm@12298
   605
    ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
paulson@14208
   606
  apply (rule LeastMI2, assumption, blast)
wenzelm@12298
   607
  apply (blast intro!: order_antisym)
wenzelm@12298
   608
  done
paulson@11451
   609
paulson@11454
   610
lemma wf_linord_ex_has_least:
paulson@14760
   611
  "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
paulson@14760
   612
    ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
wenzelm@12298
   613
  apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
paulson@14208
   614
  apply (drule_tac x = "m`Collect P" in spec, force)
wenzelm@12298
   615
  done
paulson@11454
   616
paulson@11454
   617
lemma ex_has_least_nat:
paulson@14760
   618
    "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
wenzelm@12298
   619
  apply (simp only: pred_nat_trancl_eq_le [symmetric])
wenzelm@12298
   620
  apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
paulson@16796
   621
   apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
wenzelm@12298
   622
  done
paulson@11454
   623
wenzelm@12298
   624
lemma LeastM_nat_lemma:
paulson@14760
   625
    "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
paulson@14760
   626
  apply (simp add: LeastM_def)
wenzelm@12298
   627
  apply (rule someI_ex)
wenzelm@12298
   628
  apply (erule ex_has_least_nat)
wenzelm@12298
   629
  done
paulson@11454
   630
wenzelm@45607
   631
lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1]
paulson@11454
   632
paulson@11454
   633
lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
paulson@14208
   634
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
paulson@11454
   635
paulson@11451
   636
wenzelm@60758
   637
subsection \<open>Greatest value operator\<close>
paulson@11451
   638
haftmann@35416
   639
definition
haftmann@35416
   640
  GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
paulson@14760
   641
  "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
wenzelm@12298
   642
haftmann@35416
   643
definition
haftmann@35416
   644
  Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
wenzelm@12298
   645
  "Greatest == GreatestM (%x. x)"
paulson@11451
   646
paulson@11451
   647
syntax
wenzelm@35115
   648
  "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
wenzelm@12298
   649
      ("GREATEST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   650
translations
wenzelm@35115
   651
  "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
paulson@11451
   652
paulson@11451
   653
lemma GreatestMI2:
wenzelm@12298
   654
  "P x ==> (!!y. P y ==> m y <= m x)
wenzelm@12298
   655
    ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
wenzelm@12298
   656
    ==> Q (GreatestM m P)"
paulson@14760
   657
  apply (simp add: GreatestM_def)
paulson@14208
   658
  apply (rule someI2_ex, blast, blast)
wenzelm@12298
   659
  done
paulson@11451
   660
paulson@11451
   661
lemma GreatestM_equality:
wenzelm@12298
   662
 "P k ==> (!!x. P x ==> m x <= m k)
wenzelm@12298
   663
    ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
paulson@14208
   664
  apply (rule_tac m = m in GreatestMI2, assumption, blast)
wenzelm@12298
   665
  apply (blast intro!: order_antisym)
wenzelm@12298
   666
  done
paulson@11451
   667
paulson@11451
   668
lemma Greatest_equality:
wenzelm@12298
   669
  "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
paulson@14760
   670
  apply (simp add: Greatest_def)
paulson@14208
   671
  apply (erule GreatestM_equality, blast)
wenzelm@12298
   672
  done
paulson@11451
   673
paulson@11451
   674
lemma ex_has_greatest_nat_lemma:
paulson@14760
   675
  "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
paulson@14760
   676
    ==> \<exists>y. P y & ~ (m y < m k + n)"
paulson@15251
   677
  apply (induct n, force)
wenzelm@12298
   678
  apply (force simp add: le_Suc_eq)
wenzelm@12298
   679
  done
paulson@11451
   680
wenzelm@12298
   681
lemma ex_has_greatest_nat:
paulson@14760
   682
  "P k ==> \<forall>y. P y --> m y < b
paulson@14760
   683
    ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
wenzelm@12298
   684
  apply (rule ccontr)
wenzelm@12298
   685
  apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
paulson@14208
   686
    apply (subgoal_tac [3] "m k <= b", auto)
wenzelm@12298
   687
  done
paulson@11451
   688
wenzelm@12298
   689
lemma GreatestM_nat_lemma:
paulson@14760
   690
  "P k ==> \<forall>y. P y --> m y < b
paulson@14760
   691
    ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
paulson@14760
   692
  apply (simp add: GreatestM_def)
wenzelm@12298
   693
  apply (rule someI_ex)
paulson@14208
   694
  apply (erule ex_has_greatest_nat, assumption)
wenzelm@12298
   695
  done
paulson@11451
   696
wenzelm@45607
   697
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]
paulson@11451
   698
wenzelm@12298
   699
lemma GreatestM_nat_le:
paulson@14760
   700
  "P x ==> \<forall>y. P y --> m y < b
wenzelm@12298
   701
    ==> (m x::nat) <= m (GreatestM m P)"
berghofe@21020
   702
  apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
wenzelm@12298
   703
  done
wenzelm@12298
   704
wenzelm@12298
   705
wenzelm@61799
   706
text \<open>\medskip Specialization to \<open>GREATEST\<close>.\<close>
wenzelm@12298
   707
paulson@14760
   708
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
paulson@14760
   709
  apply (simp add: Greatest_def)
paulson@14208
   710
  apply (rule GreatestM_natI, auto)
wenzelm@12298
   711
  done
paulson@11451
   712
wenzelm@12298
   713
lemma Greatest_le:
paulson@14760
   714
    "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
paulson@14760
   715
  apply (simp add: Greatest_def)
paulson@14208
   716
  apply (rule GreatestM_nat_le, auto)
wenzelm@12298
   717
  done
wenzelm@12298
   718
wenzelm@12298
   719
wenzelm@60758
   720
subsection \<open>An aside: bounded accessible part\<close>
haftmann@49948
   721
wenzelm@60758
   722
text \<open>Finite monotone eventually stable sequences\<close>
haftmann@49948
   723
haftmann@49948
   724
lemma finite_mono_remains_stable_implies_strict_prefix:
haftmann@49948
   725
  fixes f :: "nat \<Rightarrow> 'a::order"
haftmann@49948
   726
  assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
haftmann@49948
   727
  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
   728
  using assms
haftmann@49948
   729
proof -
haftmann@49948
   730
  have "\<exists>n. f n = f (Suc n)"
haftmann@49948
   731
  proof (rule ccontr)
haftmann@49948
   732
    assume "\<not> ?thesis"
haftmann@49948
   733
    then have "\<And>n. f n \<noteq> f (Suc n)" by auto
haftmann@49948
   734
    then have "\<And>n. f n < f (Suc n)"
wenzelm@60758
   735
      using  \<open>mono f\<close> by (auto simp: le_less mono_iff_le_Suc)
haftmann@49948
   736
    with lift_Suc_mono_less_iff[of f]
traytel@55811
   737
    have *: "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
traytel@55811
   738
    have "inj f"
traytel@55811
   739
    proof (intro injI)
traytel@55811
   740
      fix x y
traytel@55811
   741
      assume "f x = f y"
traytel@55811
   742
      then show "x = y" by (cases x y rule: linorder_cases) (auto dest: *)
traytel@55811
   743
    qed
wenzelm@60758
   744
    with \<open>finite (range f)\<close> have "finite (UNIV::nat set)"
haftmann@49948
   745
      by (rule finite_imageD)
haftmann@49948
   746
    then show False by simp
haftmann@49948
   747
  qed
haftmann@49948
   748
  then obtain n where n: "f n = f (Suc n)" ..
haftmann@49948
   749
  def N \<equiv> "LEAST n. f n = f (Suc n)"
haftmann@49948
   750
  have N: "f N = f (Suc N)"
haftmann@49948
   751
    unfolding N_def using n by (rule LeastI)
haftmann@49948
   752
  show ?thesis
haftmann@49948
   753
  proof (intro exI[of _ N] conjI allI impI)
haftmann@49948
   754
    fix n assume "N \<le> n"
haftmann@49948
   755
    then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
haftmann@49948
   756
    proof (induct rule: dec_induct)
haftmann@49948
   757
      case (step n) then show ?case
haftmann@49948
   758
        using eq[rule_format, of "n - 1"] N
haftmann@49948
   759
        by (cases n) (auto simp add: le_Suc_eq)
haftmann@49948
   760
    qed simp
wenzelm@60758
   761
    from this[of n] \<open>N \<le> n\<close> show "f N = f n" by auto
haftmann@49948
   762
  next
haftmann@49948
   763
    fix n m :: nat assume "m < n" "n \<le> N"
haftmann@49948
   764
    then show "f m < f n"
haftmann@49948
   765
    proof (induct rule: less_Suc_induct[consumes 1])
haftmann@49948
   766
      case (1 i)
haftmann@49948
   767
      then have "i < N" by simp
haftmann@49948
   768
      then have "f i \<noteq> f (Suc i)"
haftmann@49948
   769
        unfolding N_def by (rule not_less_Least)
wenzelm@60758
   770
      with \<open>mono f\<close> show ?case by (simp add: mono_iff_le_Suc less_le)
haftmann@49948
   771
    qed auto
haftmann@49948
   772
  qed
haftmann@49948
   773
qed
haftmann@49948
   774
haftmann@49948
   775
lemma finite_mono_strict_prefix_implies_finite_fixpoint:
haftmann@49948
   776
  fixes f :: "nat \<Rightarrow> 'a set"
haftmann@49948
   777
  assumes S: "\<And>i. f i \<subseteq> S" "finite S"
haftmann@49948
   778
    and inj: "\<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
   779
  shows "f (card S) = (\<Union>n. f n)"
haftmann@49948
   780
proof -
haftmann@49948
   781
  from inj obtain N where inj: "(\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n)" and eq: "(\<forall>n\<ge>N. f N = f n)" by auto
haftmann@49948
   782
haftmann@49948
   783
  { fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
haftmann@49948
   784
    proof (induct i)
haftmann@49948
   785
      case 0 then show ?case by simp
haftmann@49948
   786
    next
haftmann@49948
   787
      case (Suc i)
haftmann@49948
   788
      with inj[rule_format, of "Suc i" i]
haftmann@49948
   789
      have "(f i) \<subset> (f (Suc i))" by auto
haftmann@49948
   790
      moreover have "finite (f (Suc i))" using S by (rule finite_subset)
haftmann@49948
   791
      ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
haftmann@49948
   792
      with Suc show ?case using inj by auto
haftmann@49948
   793
    qed
haftmann@49948
   794
  }
haftmann@49948
   795
  then have "N \<le> card (f N)" by simp
haftmann@49948
   796
  also have "\<dots> \<le> card S" using S by (intro card_mono)
haftmann@49948
   797
  finally have "f (card S) = f N" using eq by auto
haftmann@49948
   798
  then show ?thesis using eq inj[rule_format, of N]
haftmann@49948
   799
    apply auto
haftmann@49948
   800
    apply (case_tac "n < N")
haftmann@49948
   801
    apply (auto simp: not_less)
haftmann@49948
   802
    done
haftmann@49948
   803
qed
haftmann@49948
   804
haftmann@49948
   805
wenzelm@60758
   806
subsection \<open>More on injections, bijections, and inverses\<close>
blanchet@55020
   807
blanchet@55020
   808
lemma infinite_imp_bij_betw:
blanchet@55020
   809
assumes INF: "\<not> finite A"
blanchet@55020
   810
shows "\<exists>h. bij_betw h A (A - {a})"
blanchet@55020
   811
proof(cases "a \<in> A")
blanchet@55020
   812
  assume Case1: "a \<notin> A"  hence "A - {a} = A" by blast
blanchet@55020
   813
  thus ?thesis using bij_betw_id[of A] by auto
blanchet@55020
   814
next
blanchet@55020
   815
  assume Case2: "a \<in> A"
blanchet@55020
   816
  have "\<not> finite (A - {a})" using INF by auto
blanchet@55020
   817
  with infinite_iff_countable_subset[of "A - {a}"] obtain f::"nat \<Rightarrow> 'a"
blanchet@55020
   818
  where 1: "inj f" and 2: "f ` UNIV \<le> A - {a}" by blast
blanchet@55020
   819
  obtain g where g_def: "g = (\<lambda> n. if n = 0 then a else f (Suc n))" by blast
blanchet@55020
   820
  obtain A' where A'_def: "A' = g ` UNIV" by blast
blanchet@55020
   821
  have temp: "\<forall>y. f y \<noteq> a" using 2 by blast
blanchet@55020
   822
  have 3: "inj_on g UNIV \<and> g ` UNIV \<le> A \<and> a \<in> g ` UNIV"
blanchet@55020
   823
  proof(auto simp add: Case2 g_def, unfold inj_on_def, intro ballI impI,
blanchet@55020
   824
        case_tac "x = 0", auto simp add: 2)
blanchet@55020
   825
    fix y  assume "a = (if y = 0 then a else f (Suc y))"
blanchet@55020
   826
    thus "y = 0" using temp by (case_tac "y = 0", auto)
blanchet@55020
   827
  next
blanchet@55020
   828
    fix x y
blanchet@55020
   829
    assume "f (Suc x) = (if y = 0 then a else f (Suc y))"
blanchet@55020
   830
    thus "x = y" using 1 temp unfolding inj_on_def by (case_tac "y = 0", auto)
blanchet@55020
   831
  next
blanchet@55020
   832
    fix n show "f (Suc n) \<in> A" using 2 by blast
blanchet@55020
   833
  qed
blanchet@55020
   834
  hence 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<le> A"
blanchet@55020
   835
  using inj_on_imp_bij_betw[of g] unfolding A'_def by auto
blanchet@55020
   836
  hence 5: "bij_betw (inv g) A' UNIV"
blanchet@55020
   837
  by (auto simp add: bij_betw_inv_into)
blanchet@55020
   838
  (*  *)
blanchet@55020
   839
  obtain n where "g n = a" using 3 by auto
blanchet@55020
   840
  hence 6: "bij_betw g (UNIV - {n}) (A' - {a})"
blanchet@55020
   841
  using 3 4 unfolding A'_def
blanchet@55020
   842
  by clarify (rule bij_betw_subset, auto simp: image_set_diff)
blanchet@55020
   843
  (*  *)
blanchet@55020
   844
  obtain v where v_def: "v = (\<lambda> m. if m < n then m else Suc m)" by blast
blanchet@55020
   845
  have 7: "bij_betw v UNIV (UNIV - {n})"
blanchet@55020
   846
  proof(unfold bij_betw_def inj_on_def, intro conjI, clarify)
blanchet@55020
   847
    fix m1 m2 assume "v m1 = v m2"
blanchet@55020
   848
    thus "m1 = m2"
blanchet@55020
   849
    by(case_tac "m1 < n", case_tac "m2 < n",
blanchet@55020
   850
       auto simp add: inj_on_def v_def, case_tac "m2 < n", auto)
blanchet@55020
   851
  next
blanchet@55020
   852
    show "v ` UNIV = UNIV - {n}"
blanchet@55020
   853
    proof(auto simp add: v_def)
blanchet@55020
   854
      fix m assume *: "m \<noteq> n" and **: "m \<notin> Suc ` {m'. \<not> m' < n}"
blanchet@55020
   855
      {assume "n \<le> m" with * have 71: "Suc n \<le> m" by auto
blanchet@55020
   856
       then obtain m' where 72: "m = Suc m'" using Suc_le_D by auto
blanchet@55020
   857
       with 71 have "n \<le> m'" by auto
blanchet@55020
   858
       with 72 ** have False by auto
blanchet@55020
   859
      }
blanchet@55020
   860
      thus "m < n" by force
blanchet@55020
   861
    qed
blanchet@55020
   862
  qed
blanchet@55020
   863
  (*  *)
blanchet@55020
   864
  obtain h' where h'_def: "h' = g o v o (inv g)" by blast
blanchet@55020
   865
  hence 8: "bij_betw h' A' (A' - {a})" using 5 7 6
blanchet@55020
   866
  by (auto simp add: bij_betw_trans)
blanchet@55020
   867
  (*  *)
blanchet@55020
   868
  obtain h where h_def: "h = (\<lambda> b. if b \<in> A' then h' b else b)" by blast
blanchet@55020
   869
  have "\<forall>b \<in> A'. h b = h' b" unfolding h_def by auto
blanchet@55020
   870
  hence "bij_betw h  A' (A' - {a})" using 8 bij_betw_cong[of A' h] by auto
blanchet@55020
   871
  moreover
blanchet@55020
   872
  {have "\<forall>b \<in> A - A'. h b = b" unfolding h_def by auto
blanchet@55020
   873
   hence "bij_betw h  (A - A') (A - A')"
blanchet@55020
   874
   using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto
blanchet@55020
   875
  }
blanchet@55020
   876
  moreover
blanchet@55020
   877
  have "(A' Int (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>
blanchet@55020
   878
        ((A' - {a}) Int (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"
blanchet@55020
   879
  using 4 by blast
blanchet@55020
   880
  ultimately have "bij_betw h A (A - {a})"
blanchet@55020
   881
  using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp
blanchet@55020
   882
  thus ?thesis by blast
blanchet@55020
   883
qed
blanchet@55020
   884
blanchet@55020
   885
lemma infinite_imp_bij_betw2:
blanchet@55020
   886
assumes INF: "\<not> finite A"
blanchet@55020
   887
shows "\<exists>h. bij_betw h A (A \<union> {a})"
blanchet@55020
   888
proof(cases "a \<in> A")
blanchet@55020
   889
  assume Case1: "a \<in> A"  hence "A \<union> {a} = A" by blast
blanchet@55020
   890
  thus ?thesis using bij_betw_id[of A] by auto
blanchet@55020
   891
next
blanchet@55020
   892
  let ?A' = "A \<union> {a}"
blanchet@55020
   893
  assume Case2: "a \<notin> A" hence "A = ?A' - {a}" by blast
blanchet@55020
   894
  moreover have "\<not> finite ?A'" using INF by auto
blanchet@55020
   895
  ultimately obtain f where "bij_betw f ?A' A"
blanchet@55020
   896
  using infinite_imp_bij_betw[of ?A' a] by auto
blanchet@55020
   897
  hence "bij_betw(inv_into ?A' f) A ?A'" using bij_betw_inv_into by blast
blanchet@55020
   898
  thus ?thesis by auto
blanchet@55020
   899
qed
blanchet@55020
   900
blanchet@55020
   901
lemma bij_betw_inv_into_left:
blanchet@55020
   902
assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"
blanchet@55020
   903
shows "(inv_into A f) (f a) = a"
blanchet@55020
   904
using assms unfolding bij_betw_def
blanchet@55020
   905
by clarify (rule inv_into_f_f)
blanchet@55020
   906
blanchet@55020
   907
lemma bij_betw_inv_into_right:
blanchet@55020
   908
assumes "bij_betw f A A'" "a' \<in> A'"
blanchet@55020
   909
shows "f(inv_into A f a') = a'"
blanchet@55020
   910
using assms unfolding bij_betw_def using f_inv_into_f by force
blanchet@55020
   911
blanchet@55020
   912
lemma bij_betw_inv_into_subset:
blanchet@55020
   913
assumes BIJ: "bij_betw f A A'" and
blanchet@55020
   914
        SUB: "B \<le> A" and IM: "f ` B = B'"
blanchet@55020
   915
shows "bij_betw (inv_into A f) B' B"
blanchet@55020
   916
using assms unfolding bij_betw_def
blanchet@55020
   917
by (auto intro: inj_on_inv_into)
blanchet@55020
   918
blanchet@55020
   919
wenzelm@60758
   920
subsection \<open>Specification package -- Hilbertized version\<close>
wenzelm@17893
   921
wenzelm@17893
   922
lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
wenzelm@17893
   923
  by (simp only: someI_ex)
wenzelm@17893
   924
wenzelm@48891
   925
ML_file "Tools/choice_specification.ML"
skalberg@14115
   926
paulson@11451
   927
end