author  wenzelm 
Fri, 26 Jun 2015 10:20:33 +0200  
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parent 59094  9ced35b4a2a9 
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permissions  rwrr 
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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 {* Hilbert's EpsilonOperator and the Axiom of Choice *} 
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theory Hilbert_Choice 
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imports Nat Wellfounded 
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keywords "specification" :: thy_goal 
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begin 
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subsection {* Hilbert's epsilon *} 

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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 {* 
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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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*}  {* to avoid etacontraction of body *} 

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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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39 

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subsection {*Hilbert's Epsilonoperator*} 

41 

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text{*Easier to apply than @{text someI} if the witness comes from an 

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existential formula*} 

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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{*Easier to apply than @{text someI} because the conclusion has only one 

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occurrence of @{term P}.*} 

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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) 

53 

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text{*Easier to apply than @{text someI2} if the witness comes from an 

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existential formula*} 

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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 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) 

62 

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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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83 

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subsection{*Axiom of Choice, Proved Using the Description Operator*} 

85 

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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) 

91 

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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) 

94 

95 
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 {*Function Inverse*} 
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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{*But is it useful?*} 

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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: image_eq_UN surj_f_inv_f) 

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lemma image_inv_f_f: "inj f ==> inv f ` (f ` A) = A" 
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by (simp add: image_eq_UN) 

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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 

33057  285 
from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def) 
31380  286 
thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast 
287 
qed 

288 
ultimately show "finite (UNIV :: 'a set)" by simp 

289 
qed 

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text {* 
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Every infinite set contains a countable subset. More precisely we 
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show that a set @{text S} is infinite if and only if there exists an 
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injective function from the naturals into @{text S}. 
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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 
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infinite set @{text S}. The idea is to construct a sequence of 
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nonempty and infinite subsets of @{text S} obtained by successively 
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removing elements of @{text S}. 
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*} 
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lemma infinite_countable_subset: 
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assumes inf: "\<not> finite (S::'a set)" 
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shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S" 
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 {* Courtesy of Stephan Merz *} 
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proof  
55415  308 
def Sseq \<equiv> "rec_nat S (\<lambda>n T. T  {SOME e. e \<in> T})" 
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

309 
def pick \<equiv> "\<lambda>n. (SOME e. e \<in> Sseq n)" 
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

310 
{ fix n have "Sseq n \<subseteq> S" "\<not> finite (Sseq n)" by (induct n) (auto simp add: Sseq_def inf) } 
55811  311 
moreover then have *: "\<And>n. pick n \<in> Sseq n" 
312 
unfolding pick_def by (subst (asm) finite.simps) (auto simp add: ex_in_conv intro: someI_ex) 

54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

313 
ultimately have "range pick \<subseteq> S" by auto 
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

314 
moreover 
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

315 
{ fix n m 
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

316 
have "pick n \<notin> Sseq (n + Suc m)" by (induct m) (auto simp add: Sseq_def pick_def) 
55811  317 
with * have "pick n \<noteq> pick (n + Suc m)" by auto 
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

318 
} 
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

319 
then have "inj pick" by (intro linorder_injI) (auto simp add: less_iff_Suc_add) 
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

320 
ultimately show ?thesis by blast 
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

321 
qed 
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

322 

9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

323 
lemma infinite_iff_countable_subset: "\<not> finite S \<longleftrightarrow> (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)" 
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

324 
 {* Courtesy of Stephan Merz *} 
55811  325 
using finite_imageD finite_subset infinite_UNIV_char_0 infinite_countable_subset by auto 
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54295
diff
changeset

326 

40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

327 
lemma image_inv_into_cancel: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

328 
assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

329 
shows "f `((inv_into A f)`B') = B'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

330 
using assms 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

331 
proof (auto simp add: f_inv_into_f) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

332 
let ?f' = "(inv_into A f)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

333 
fix a' assume *: "a' \<in> B'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

334 
then have "a' \<in> A'" using SUB by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

335 
then have "a' = f (?f' a')" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

336 
using SURJ by (auto simp add: f_inv_into_f) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

337 
then show "a' \<in> f ` (?f' ` B')" using * by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

338 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

339 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

340 
lemma inv_into_inv_into_eq: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

341 
assumes "bij_betw f A A'" "a \<in> A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

342 
shows "inv_into A' (inv_into A f) a = f a" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

343 
proof  
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

344 
let ?f' = "inv_into A f" let ?f'' = "inv_into A' ?f'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

345 
have 1: "bij_betw ?f' A' A" using assms 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

346 
by (auto simp add: bij_betw_inv_into) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

347 
obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

348 
using 1 `a \<in> A` unfolding bij_betw_def by force 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

349 
hence "?f'' a = a'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

350 
using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

351 
moreover have "f a = a'" using assms 2 3 
44921  352 
by (auto simp add: bij_betw_def) 
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

353 
ultimately show "?f'' a = f a" by simp 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

354 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

355 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

356 
lemma inj_on_iff_surj: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

357 
assumes "A \<noteq> {}" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

358 
shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

359 
proof safe 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

360 
fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

361 
let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'" let ?csi = "\<lambda>a. a \<in> A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

362 
let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

363 
have "?g ` A' = A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

364 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

365 
show "?g ` A' \<le> A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

366 
proof clarify 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

367 
fix a' assume *: "a' \<in> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

368 
show "?g a' \<in> A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

369 
proof cases 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

370 
assume Case1: "a' \<in> f ` A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

371 
then obtain a where "?phi a' a" by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

372 
hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

373 
with Case1 show ?thesis by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

374 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

375 
assume Case2: "a' \<notin> f ` A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

376 
hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

377 
with Case2 show ?thesis by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

378 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

379 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

380 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

381 
show "A \<le> ?g ` A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

382 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

383 
{fix a assume *: "a \<in> A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

384 
let ?b = "SOME aa. ?phi (f a) aa" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

385 
have "?phi (f a) a" using * by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

386 
hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

387 
hence "?g(f a) = ?b" using * by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

388 
moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

389 
ultimately have "?g(f a) = a" by simp 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

390 
with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

391 
} 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

392 
thus ?thesis by force 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

393 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

394 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

395 
thus "\<exists>g. g ` A' = A" by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

396 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

397 
fix g let ?f = "inv_into A' g" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

398 
have "inj_on ?f (g ` A')" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

399 
by (auto simp add: inj_on_inv_into) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

400 
moreover 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

401 
{fix a' assume *: "a' \<in> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

402 
let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

403 
have "?phi a'" using * by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

404 
hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

405 
hence "?f(g a') \<in> A'" unfolding inv_into_def by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

406 
} 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

407 
ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

408 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

409 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

410 
lemma Ex_inj_on_UNION_Sigma: 
60585  411 
"\<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))" 
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

412 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

413 
let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

414 
let ?sm = "\<lambda> a. SOME i. ?phi a i" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

415 
let ?f = "\<lambda>a. (?sm a, a)" 
60585  416 
have "inj_on ?f (\<Union>i \<in> I. A i)" unfolding inj_on_def by auto 
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

417 
moreover 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

418 
{ { fix i a assume "i \<in> I" and "a \<in> A i" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

419 
hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

420 
} 
60585  421 
hence "?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto 
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

422 
} 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

423 
ultimately 
60585  424 
show "inj_on ?f (\<Union>i \<in> I. A i) \<and> ?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)" 
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

425 
by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

426 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

427 

56608  428 
lemma inv_unique_comp: 
429 
assumes fg: "f \<circ> g = id" 

430 
and gf: "g \<circ> f = id" 

431 
shows "inv f = g" 

432 
using fg gf inv_equality[of g f] by (auto simp add: fun_eq_iff) 

433 

434 

40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

435 
subsection {* The CantorBernstein Theorem *} 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

436 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

437 
lemma Cantor_Bernstein_aux: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

438 
shows "\<exists>A' h. A' \<le> A \<and> 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

439 
(\<forall>a \<in> A'. a \<notin> g`(B  f ` A')) \<and> 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

440 
(\<forall>a \<in> A'. h a = f a) \<and> 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

441 
(\<forall>a \<in> A  A'. h a \<in> B  (f ` A') \<and> a = g(h a))" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

442 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

443 
obtain H where H_def: "H = (\<lambda> A'. A  (g`(B  (f ` A'))))" by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

444 
have 0: "mono H" unfolding mono_def H_def by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

445 
then obtain A' where 1: "H A' = A'" using lfp_unfold by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

446 
hence 2: "A' = A  (g`(B  (f ` A')))" unfolding H_def by simp 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

447 
hence 3: "A' \<le> A" by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

448 
have 4: "\<forall>a \<in> A'. a \<notin> g`(B  f ` A')" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

449 
using 2 by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

450 
have 5: "\<forall>a \<in> A  A'. \<exists>b \<in> B  (f ` A'). a = g b" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

451 
using 2 by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

452 
(* *) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

453 
obtain h where h_def: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

454 
"h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B  (f ` A') \<and> a = g b))" by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

455 
hence "\<forall>a \<in> A'. h a = f a" by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

456 
moreover 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

457 
have "\<forall>a \<in> A  A'. h a \<in> B  (f ` A') \<and> a = g(h a)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

458 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

459 
fix a assume *: "a \<in> A  A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

460 
let ?phi = "\<lambda> b. b \<in> B  (f ` A') \<and> a = g b" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

461 
have "h a = (SOME b. ?phi b)" using h_def * by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

462 
moreover have "\<exists>b. ?phi b" using 5 * by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

463 
ultimately show "?phi (h a)" using someI_ex[of ?phi] by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

464 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

465 
ultimately show ?thesis using 3 4 by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

466 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

467 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

468 
theorem Cantor_Bernstein: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

469 
assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

470 
INJ2: "inj_on g B" and SUB2: "g ` B \<le> A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

471 
shows "\<exists>h. bij_betw h A B" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

472 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

473 
obtain A' and h where 0: "A' \<le> A" and 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

474 
1: "\<forall>a \<in> A'. a \<notin> g`(B  f ` A')" and 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

475 
2: "\<forall>a \<in> A'. h a = f a" and 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

476 
3: "\<forall>a \<in> A  A'. h a \<in> B  (f ` A') \<and> a = g(h a)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

477 
using Cantor_Bernstein_aux[of A g B f] by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

478 
have "inj_on h A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

479 
proof (intro inj_onI) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

480 
fix a1 a2 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

481 
assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

482 
show "a1 = a2" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

483 
proof(cases "a1 \<in> A'") 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

484 
assume Case1: "a1 \<in> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

485 
show ?thesis 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

486 
proof(cases "a2 \<in> A'") 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

487 
assume Case11: "a2 \<in> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

488 
hence "f a1 = f a2" using Case1 2 6 by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

489 
thus ?thesis using INJ1 Case1 Case11 0 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

490 
unfolding inj_on_def by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

491 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

492 
assume Case12: "a2 \<notin> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

493 
hence False using 3 5 2 6 Case1 by force 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

494 
thus ?thesis by simp 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

495 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

496 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

497 
assume Case2: "a1 \<notin> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

498 
show ?thesis 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

499 
proof(cases "a2 \<in> A'") 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

500 
assume Case21: "a2 \<in> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

501 
hence False using 3 4 2 6 Case2 by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

502 
thus ?thesis by simp 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

503 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

504 
assume Case22: "a2 \<notin> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

505 
hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

506 
thus ?thesis using 6 by simp 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

507 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

508 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

509 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

510 
(* *) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

511 
moreover 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

512 
have "h ` A = B" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

513 
proof safe 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

514 
fix a assume "a \<in> A" 
47988  515 
thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto 
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

516 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

517 
fix b assume *: "b \<in> B" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

518 
show "b \<in> h ` A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

519 
proof(cases "b \<in> f ` A'") 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

520 
assume Case1: "b \<in> f ` A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

521 
then obtain a where "a \<in> A' \<and> b = f a" by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

522 
thus ?thesis using 2 0 by force 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

523 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

524 
assume Case2: "b \<notin> f ` A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

525 
hence "g b \<notin> A'" using 1 * by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

526 
hence 4: "g b \<in> A  A'" using * SUB2 by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

527 
hence "h(g b) \<in> B \<and> g(h(g b)) = g b" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

528 
using 3 by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

529 
hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

530 
thus ?thesis using 4 by force 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

531 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

532 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

533 
(* *) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

534 
ultimately show ?thesis unfolding bij_betw_def by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

535 
qed 
14760  536 

537 
subsection {*Other Consequences of Hilbert's Epsilon*} 

538 

539 
text {*Hilbert's Epsilon and the @{term split} Operator*} 

540 

541 
text{*Looping simprule*} 

542 
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))" 

26347  543 
by simp 
14760  544 

545 
lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))" 

26347  546 
by (simp add: split_def) 
14760  547 

548 
lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)" 

26347  549 
by blast 
14760  550 

551 

552 
text{*A relation is wellfounded iff it has no infinite descending chain*} 

553 
lemma wf_iff_no_infinite_down_chain: 

554 
"wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))" 

555 
apply (simp only: wf_eq_minimal) 

556 
apply (rule iffI) 

557 
apply (rule notI) 

558 
apply (erule exE) 

559 
apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast) 

560 
apply (erule contrapos_np, simp, clarify) 

55415  561 
apply (subgoal_tac "\<forall>n. rec_nat x (%i y. @z. z:Q & (z,y) :r) n \<in> Q") 
562 
apply (rule_tac x = "rec_nat x (%i y. @z. z:Q & (z,y) :r)" in exI) 

14760  563 
apply (rule allI, simp) 
564 
apply (rule someI2_ex, blast, blast) 

565 
apply (rule allI) 

566 
apply (induct_tac "n", simp_all) 

567 
apply (rule someI2_ex, blast+) 

568 
done 

569 

27760  570 
lemma wf_no_infinite_down_chainE: 
571 
assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r" 

572 
using `wf r` wf_iff_no_infinite_down_chain[of r] by blast 

573 

574 

14760  575 
text{*A dynamicallyscoped fact for TFL *} 
12298  576 
lemma tfl_some: "\<forall>P x. P x > P (Eps P)" 
577 
by (blast intro: someI) 

11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

578 

12298  579 

580 
subsection {* Least value operator *} 

11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

581 

35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset

582 
definition 
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset

583 
LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where 
14760  584 
"LeastM m P == SOME x. P x & (\<forall>y. P y > m x <= m y)" 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

585 

8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

586 
syntax 
12298  587 
"_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a" ("LEAST _ WRT _. _" [0, 4, 10] 10) 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

588 
translations 
35115  589 
"LEAST x WRT m. P" == "CONST LeastM m (%x. P)" 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

590 

8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

591 
lemma LeastMI2: 
12298  592 
"P x ==> (!!y. P y ==> m x <= m y) 
593 
==> (!!x. P x ==> \<forall>y. P y > m x \<le> m y ==> Q x) 

594 
==> Q (LeastM m P)" 

14760  595 
apply (simp add: LeastM_def) 
14208  596 
apply (rule someI2_ex, blast, blast) 
12298  597 
done 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

598 

8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

599 
lemma LeastM_equality: 
12298  600 
"P k ==> (!!x. P x ==> m k <= m x) 
601 
==> m (LEAST x WRT m. P x) = (m k::'a::order)" 

14208  602 
apply (rule LeastMI2, assumption, blast) 
12298  603 
apply (blast intro!: order_antisym) 
604 
done 

11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

605 

11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset

606 
lemma wf_linord_ex_has_least: 
14760  607 
"wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k 
608 
==> \<exists>x. P x & (!y. P y > (m x,m y):r^*)" 

12298  609 
apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]]) 
14208  610 
apply (drule_tac x = "m`Collect P" in spec, force) 
12298  611 
done 
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset

612 

7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset

613 
lemma ex_has_least_nat: 
14760  614 
"P k ==> \<exists>x. P x & (\<forall>y. P y > m x <= (m y::nat))" 
12298  615 
apply (simp only: pred_nat_trancl_eq_le [symmetric]) 
616 
apply (rule wf_pred_nat [THEN wf_linord_ex_has_least]) 

16796  617 
apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption) 
12298  618 
done 
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset

619 

12298  620 
lemma LeastM_nat_lemma: 
14760  621 
"P k ==> P (LeastM m P) & (\<forall>y. P y > m (LeastM m P) <= (m y::nat))" 
622 
apply (simp add: LeastM_def) 

12298  623 
apply (rule someI_ex) 
624 
apply (erule ex_has_least_nat) 

625 
done 

11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset

626 

45607  627 
lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1] 
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset

628 

7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset

629 
lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)" 
14208  630 
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption) 
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset

631 

11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

632 

12298  633 
subsection {* Greatest value operator *} 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

634 

35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset

635 
definition 
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset

636 
GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where 
14760  637 
"GreatestM m P == SOME x. P x & (\<forall>y. P y > m y <= m x)" 
12298  638 

35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset

639 
definition 
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset

640 
Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where 
12298  641 
"Greatest == GreatestM (%x. x)" 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

642 

8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

643 
syntax 
35115  644 
"_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a" 
12298  645 
("GREATEST _ WRT _. _" [0, 4, 10] 10) 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

646 
translations 
35115  647 
"GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)" 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

648 

8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

649 
lemma GreatestMI2: 
12298  650 
"P x ==> (!!y. P y ==> m y <= m x) 
651 
==> (!!x. P x ==> \<forall>y. P y > m y \<le> m x ==> Q x) 

652 
==> Q (GreatestM m P)" 

14760  653 
apply (simp add: GreatestM_def) 
14208  654 
apply (rule someI2_ex, blast, blast) 
12298  655 
done 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

656 

8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

657 
lemma GreatestM_equality: 
12298  658 
"P k ==> (!!x. P x ==> m x <= m k) 
659 
==> m (GREATEST x WRT m. P x) = (m k::'a::order)" 

14208  660 
apply (rule_tac m = m in GreatestMI2, assumption, blast) 
12298  661 
apply (blast intro!: order_antisym) 
662 
done 

11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

663 

8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

664 
lemma Greatest_equality: 
12298  665 
"P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k" 
14760  666 
apply (simp add: Greatest_def) 
14208  667 
apply (erule GreatestM_equality, blast) 
12298  668 
done 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

669 

8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

670 
lemma ex_has_greatest_nat_lemma: 
14760  671 
"P k ==> \<forall>x. P x > (\<exists>y. P y & ~ ((m y::nat) <= m x)) 
672 
==> \<exists>y. P y & ~ (m y < m k + n)" 

15251  673 
apply (induct n, force) 
12298  674 
apply (force simp add: le_Suc_eq) 
675 
done 

11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

676 

12298  677 
lemma ex_has_greatest_nat: 
14760  678 
"P k ==> \<forall>y. P y > m y < b 
679 
==> \<exists>x. P x & (\<forall>y. P y > (m y::nat) <= m x)" 

12298  680 
apply (rule ccontr) 
681 
apply (cut_tac P = P and n = "b  m k" in ex_has_greatest_nat_lemma) 

14208  682 
apply (subgoal_tac [3] "m k <= b", auto) 
12298  683 
done 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

684 

12298  685 
lemma GreatestM_nat_lemma: 
14760  686 
"P k ==> \<forall>y. P y > m y < b 
687 
==> P (GreatestM m P) & (\<forall>y. P y > (m y::nat) <= m (GreatestM m P))" 

688 
apply (simp add: GreatestM_def) 

12298  689 
apply (rule someI_ex) 
14208  690 
apply (erule ex_has_greatest_nat, assumption) 
12298  691 
done 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

692 

45607  693 
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1] 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

694 

12298  695 
lemma GreatestM_nat_le: 
14760  696 
"P x ==> \<forall>y. P y > m y < b 
12298  697 
==> (m x::nat) <= m (GreatestM m P)" 
21020  698 
apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P]) 
12298  699 
done 
700 

701 

702 
text {* \medskip Specialization to @{text GREATEST}. *} 

703 

14760  704 
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y > y < b ==> P (GREATEST x. P x)" 
705 
apply (simp add: Greatest_def) 

14208  706 
apply (rule GreatestM_natI, auto) 
12298  707 
done 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

708 

12298  709 
lemma Greatest_le: 
14760  710 
"P x ==> \<forall>y. P y > y < b ==> (x::nat) <= (GREATEST x. P x)" 
711 
apply (simp add: Greatest_def) 

14208  712 
apply (rule GreatestM_nat_le, auto) 
12298  713 
done 
714 

715 

49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

716 
subsection {* An aside: bounded accessible part *} 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

717 

744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

718 
text {* Finite monotone eventually stable sequences *} 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

719 

744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

720 
lemma finite_mono_remains_stable_implies_strict_prefix: 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

721 
fixes f :: "nat \<Rightarrow> 'a::order" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

722 
assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

723 
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)" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

724 
using assms 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

725 
proof  
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

726 
have "\<exists>n. f n = f (Suc n)" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

727 
proof (rule ccontr) 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

728 
assume "\<not> ?thesis" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

729 
then have "\<And>n. f n \<noteq> f (Suc n)" by auto 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

730 
then have "\<And>n. f n < f (Suc n)" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

731 
using `mono f` by (auto simp: le_less mono_iff_le_Suc) 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

732 
with lift_Suc_mono_less_iff[of f] 
55811  733 
have *: "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto 
734 
have "inj f" 

735 
proof (intro injI) 

736 
fix x y 

737 
assume "f x = f y" 

738 
then show "x = y" by (cases x y rule: linorder_cases) (auto dest: *) 

739 
qed 

49948
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

740 
with `finite (range f)` have "finite (UNIV::nat set)" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

741 
by (rule finite_imageD) 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

742 
then show False by simp 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

743 
qed 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

744 
then obtain n where n: "f n = f (Suc n)" .. 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

745 
def N \<equiv> "LEAST n. f n = f (Suc n)" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

746 
have N: "f N = f (Suc N)" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

747 
unfolding N_def using n by (rule LeastI) 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

748 
show ?thesis 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

749 
proof (intro exI[of _ N] conjI allI impI) 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

750 
fix n assume "N \<le> n" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

751 
then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

752 
proof (induct rule: dec_induct) 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

753 
case (step n) then show ?case 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

754 
using eq[rule_format, of "n  1"] N 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

755 
by (cases n) (auto simp add: le_Suc_eq) 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

756 
qed simp 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

757 
from this[of n] `N \<le> n` show "f N = f n" by auto 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

758 
next 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

759 
fix n m :: nat assume "m < n" "n \<le> N" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

760 
then show "f m < f n" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

761 
proof (induct rule: less_Suc_induct[consumes 1]) 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

762 
case (1 i) 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

763 
then have "i < N" by simp 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

764 
then have "f i \<noteq> f (Suc i)" 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

765 
unfolding N_def by (rule not_less_Least) 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

766 
with `mono f` show ?case by (simp add: mono_iff_le_Suc less_le) 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

767 
qed auto 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

768 
qed 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
diff
changeset

769 
qed 
744934b818c7
moved quite generic material from theory Enum to more appropriate places
haftmann
parents:
49739
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770 

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771 
lemma finite_mono_strict_prefix_implies_finite_fixpoint: 
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772 
fixes f :: "nat \<Rightarrow> 'a set" 
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diff
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773 
assumes S: "\<And>i. f i \<subseteq> S" "finite S" 
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774 
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)" 
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775 
shows "f (card S) = (\<Union>n. f n)" 
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776 
proof  
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777 
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 
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778 

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779 
{ fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)" 
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780 
proof (induct i) 
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781 
case 0 then show ?case by simp 
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782 
next 
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783 
case (Suc i) 
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784 
with inj[rule_format, of "Suc i" i] 
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785 
have "(f i) \<subset> (f (Suc i))" by auto 
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diff
changeset

786 
moreover have "finite (f (Suc i))" using S by (rule finite_subset) 
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diff
changeset

787 
ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono) 
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788 
with Suc show ?case using inj by auto 
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789 
qed 
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790 
} 
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791 
then have "N \<le> card (f N)" by simp 
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diff
changeset

792 
also have "\<dots> \<le> card S" using S by (intro card_mono) 
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diff
changeset

793 
finally have "f (card S) = f N" using eq by auto 
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changeset

794 
then show ?thesis using eq inj[rule_format, of N] 
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diff
changeset

795 
apply auto 
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changeset

796 
apply (case_tac "n < N") 
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diff
changeset

797 
apply (auto simp: not_less) 
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changeset

798 
done 
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799 
qed 
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diff
changeset

800 

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801 

55020  802 
subsection {* More on injections, bijections, and inverses *} 
803 

804 
lemma infinite_imp_bij_betw: 

805 
assumes INF: "\<not> finite A" 

806 
shows "\<exists>h. bij_betw h A (A  {a})" 

807 
proof(cases "a \<in> A") 

808 
assume Case1: "a \<notin> A" hence "A  {a} = A" by blast 

809 
thus ?thesis using bij_betw_id[of A] by auto 

810 
next 

811 
assume Case2: "a \<in> A" 

812 
have "\<not> finite (A  {a})" using INF by auto 

813 
with infinite_iff_countable_subset[of "A  {a}"] obtain f::"nat \<Rightarrow> 'a" 

814 
where 1: "inj f" and 2: "f ` UNIV \<le> A  {a}" by blast 

815 
obtain g where g_def: "g = (\<lambda> n. if n = 0 then a else f (Suc n))" by blast 

816 
obtain A' where A'_def: "A' = g ` UNIV" by blast 

817 
have temp: "\<forall>y. f y \<noteq> a" using 2 by blast 

818 
have 3: "inj_on g UNIV \<and> g ` UNIV \<le> A \<and> a \<in> g ` UNIV" 

819 
proof(auto simp add: Case2 g_def, unfold inj_on_def, intro ballI impI, 

820 
case_tac "x = 0", auto simp add: 2) 

821 
fix y assume "a = (if y = 0 then a else f (Suc y))" 

822 
thus "y = 0" using temp by (case_tac "y = 0", auto) 

823 
next 

824 
fix x y 

825 
assume "f (Suc x) = (if y = 0 then a else f (Suc y))" 

826 
thus "x = y" using 1 temp unfolding inj_on_def by (case_tac "y = 0", auto) 

827 
next 

828 
fix n show "f (Suc n) \<in> A" using 2 by blast 

829 
qed 

830 
hence 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<le> A" 

831 
using inj_on_imp_bij_betw[of g] unfolding A'_def by auto 

832 
hence 5: "bij_betw (inv g) A' UNIV" 

833 
by (auto simp add: bij_betw_inv_into) 

834 
(* *) 

835 
obtain n where "g n = a" using 3 by auto 

836 
hence 6: "bij_betw g (UNIV  {n}) (A'  {a})" 

837 
using 3 4 unfolding A'_def 

838 
by clarify (rule bij_betw_subset, auto simp: image_set_diff) 

839 
(* *) 

840 
obtain v where v_def: "v = (\<lambda> m. if m < n then m else Suc m)" by blast 

841 
have 7: "bij_betw v UNIV (UNIV  {n})" 

842 
proof(unfold bij_betw_def inj_on_def, intro conjI, clarify) 

843 
fix m1 m2 assume "v m1 = v m2" 

844 
thus "m1 = m2" 

845 
by(case_tac "m1 < n", case_tac "m2 < n", 

846 
auto simp add: inj_on_def v_def, case_tac "m2 < n", auto) 

847 
next 

848 
show "v ` UNIV = UNIV  {n}" 

849 
proof(auto simp add: v_def) 

850 
fix m assume *: "m \<noteq> n" and **: "m \<notin> Suc ` {m'. \<not> m' < n}" 

851 
{assume "n \<le> m" with * have 71: "Suc n \<le> m" by auto 

852 
then obtain m' where 72: "m = Suc m'" using Suc_le_D by auto 

853 
with 71 have "n \<le> m'" by auto 

854 
with 72 ** have False by auto 

855 
} 

856 
thus "m < n" by force 

857 
qed 

858 
qed 

859 
(* *) 

860 
obtain h' where h'_def: "h' = g o v o (inv g)" by blast 

861 
hence 8: "bij_betw h' A' (A'  {a})" using 5 7 6 

862 
by (auto simp add: bij_betw_trans) 

863 
(* *) 

864 
obtain h where h_def: "h = (\<lambda> b. if b \<in> A' then h' b else b)" by blast 

865 
have "\<forall>b \<in> A'. h b = h' b" unfolding h_def by auto 

866 
hence "bij_betw h A' (A'  {a})" using 8 bij_betw_cong[of A' h] by auto 

867 
moreover 

868 
{have "\<forall>b \<in> A  A'. h b = b" unfolding h_def by auto 

869 
hence "bij_betw h (A  A') (A  A')" 

870 
using bij_betw_cong[of "A  A'" h id] bij_betw_id[of "A  A'"] by auto 

871 
} 

872 
moreover 

873 
have "(A' Int (A  A') = {} \<and> A' \<union> (A  A') = A) \<and> 

874 
((A'  {a}) Int (A  A') = {} \<and> (A'  {a}) \<union> (A  A') = A  {a})" 

875 
using 4 by blast 

876 
ultimately have "bij_betw h A (A  {a})" 

877 
using bij_betw_combine[of h A' "A'  {a}" "A  A'" "A  A'"] by simp 

878 
thus ?thesis by blast 

879 
qed 

880 

881 
lemma infinite_imp_bij_betw2: 

882 
assumes INF: "\<not> finite A" 

883 
shows "\<exists>h. bij_betw h A (A \<union> {a})" 

884 
proof(cases "a \<in> A") 

885 
assume Case1: "a \<in> A" hence "A \<union> {a} = A" by blast 

886 
thus ?thesis using bij_betw_id[of A] by auto 

887 
next 

888 
let ?A' = "A \<union> {a}" 

889 
assume Case2: "a \<notin> A" hence "A = ?A'  {a}" by blast 

890 
moreover have "\<not> finite ?A'" using INF by auto 

891 
ultimately obtain f where "bij_betw f ?A' A" 

892 
using infinite_imp_bij_betw[of ?A' a] by auto 

893 
hence "bij_betw(inv_into ?A' f) A ?A'" using bij_betw_inv_into by blast 

894 
thus ?thesis by auto 

895 
qed 

896 

897 
lemma bij_betw_inv_into_left: 

898 
assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A" 

899 
shows "(inv_into A f) (f a) = a" 

900 
using assms unfolding bij_betw_def 

901 
by clarify (rule inv_into_f_f) 

902 

903 
lemma bij_betw_inv_into_right: 

904 
assumes "bij_betw f A A'" "a' \<in> A'" 

905 
shows "f(inv_into A f a') = a'" 

906 
using assms unfolding bij_betw_def using f_inv_into_f by force 

907 

908 
lemma bij_betw_inv_into_subset: 

909 
assumes BIJ: "bij_betw f A A'" and 

910 
SUB: "B \<le> A" and IM: "f ` B = B'" 

911 
shows "bij_betw (inv_into A f) B' B" 

912 
using assms unfolding bij_betw_def 

913 
by (auto intro: inj_on_inv_into) 

914 

915 

17893
aef5a6d11c2a
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17702
diff
changeset

916 
subsection {* Specification package  Hilbertized version *} 
aef5a6d11c2a
added lemma exE_some (from specification_package.ML);
wenzelm
parents:
17702
diff
changeset

917 

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diff
changeset

918 
lemma exE_some: "[ Ex P ; c == Eps P ] ==> P c" 
aef5a6d11c2a
added lemma exE_some (from specification_package.ML);
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parents:
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diff
changeset

919 
by (simp only: someI_ex) 
aef5a6d11c2a
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17702
diff
changeset

920 

48891  921 
ML_file "Tools/choice_specification.ML" 
14115  922 

11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

923 
end 