author  huffman 
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permissions  rwrr 
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(* Title: HOL/Hilbert_Choice.thy 
32988  2 
Author: Lawrence C Paulson, Tobias Nipkow 
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Copyright 2001 University of Cambridge 
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*) 
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header {* Hilbert's EpsilonOperator and the Axiom of Choice *} 
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theory Hilbert_Choice 
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imports Nat Wellfounded Plain 
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uses ("Tools/choice_specification.ML") 
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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 [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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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) 

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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) 

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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_eq_trivial [simp]: "(SOME y. y=x) = x" 

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apply (rule some_equality) 

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apply (rule refl, assumption) 

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done 

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lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x" 

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apply (rule some_equality) 

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apply (rule refl) 

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apply (erule sym) 

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done 

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

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lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)" 
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by (fast elim: someI) 
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lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)" 

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by (fast elim: someI) 

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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: o_assoc[symmetric]) 

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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) 

32988  217 
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 (auto simp add: image_def) 

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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)" 

248 
proof  

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from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2) 

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with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)" 

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by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff) 

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then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set)  Suc (Suc 0)" by auto 

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then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq) 

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from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI) 

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moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)" 

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proof (rule UNIV_eq_I) 

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fix x :: 'a 

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

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

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qed 

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lemma image_inv_into_cancel: 
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assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'" 
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shows "f `((inv_into A f)`B') = B'" 
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using assms 
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proof (auto simp add: f_inv_into_f) 
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let ?f' = "(inv_into A f)" 
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fix a' assume *: "a' \<in> B'" 
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then have "a' \<in> A'" using SUB by auto 
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then have "a' = f (?f' a')" 
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using SURJ by (auto simp add: f_inv_into_f) 
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then show "a' \<in> f ` (?f' ` B')" using * by blast 
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qed 
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lemma inv_into_inv_into_eq: 
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assumes "bij_betw f A A'" "a \<in> A" 
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shows "inv_into A' (inv_into A f) a = f a" 
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proof  
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let ?f' = "inv_into A f" let ?f'' = "inv_into A' ?f'" 
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have 1: "bij_betw ?f' A' A" using assms 
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by (auto simp add: bij_betw_inv_into) 
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obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a" 
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using 1 `a \<in> A` unfolding bij_betw_def by force 
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hence "?f'' a = a'" 
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using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def) 
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moreover have "f a = a'" using assms 2 3 
44921  289 
by (auto simp add: bij_betw_def) 
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ultimately show "?f'' a = f a" by simp 
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qed 
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lemma inj_on_iff_surj: 
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assumes "A \<noteq> {}" 
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shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)" 
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proof safe 
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fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'" 
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Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

298 
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

299 
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

300 
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

301 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

302 
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

303 
proof clarify 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

304 
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

305 
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

306 
proof cases 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

307 
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

308 
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

309 
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

310 
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

311 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

312 
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

313 
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

314 
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

315 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

316 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

317 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

318 
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

319 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

320 
{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

321 
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

322 
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

323 
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

324 
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

325 
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

326 
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

327 
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

328 
} 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

329 
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

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

331 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

332 
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

333 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

334 
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

335 
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

336 
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

337 
moreover 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

338 
{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

339 
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

340 
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

341 
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

342 
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

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

344 
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

345 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

346 

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

347 
lemma Ex_inj_on_UNION_Sigma: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

348 
"\<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))" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

349 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

350 
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

351 
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

352 
let ?f = "\<lambda>a. (?sm a, a)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

353 
have "inj_on ?f (\<Union> i \<in> I. A i)" 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

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

355 
{ { 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

356 
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

357 
} 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

358 
hence "?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. 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

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

360 
ultimately 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

361 
show "inj_on ?f (\<Union> i \<in> I. A i) \<and> ?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

362 
by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

363 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

364 

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

365 
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

366 

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

367 
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

368 
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

369 
(\<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

370 
(\<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

371 
(\<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

372 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

373 
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

374 
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

375 
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

376 
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

377 
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

378 
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

379 
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

380 
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

381 
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

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

383 
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

384 
"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

385 
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

386 
moreover 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

387 
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

388 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

389 
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

390 
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

391 
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

392 
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

393 
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

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

395 
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

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

397 

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

398 
theorem Cantor_Bernstein: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

399 
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

400 
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

401 
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

402 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

403 
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

404 
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

405 
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

406 
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

407 
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

408 
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

409 
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

410 
fix a1 a2 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

411 
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

412 
show "a1 = a2" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

413 
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

414 
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

415 
show ?thesis 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

416 
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

417 
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

418 
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

419 
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

420 
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

421 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

422 
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

423 
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

424 
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

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

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

427 
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

428 
show ?thesis 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

429 
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

430 
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

431 
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

432 
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

433 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

434 
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

435 
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

436 
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

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

438 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

439 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

440 
(* *) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

441 
moreover 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

442 
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

443 
proof safe 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

444 
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

445 
thus "h a \<in> B" using SUB1 2 3 by (case_tac "a \<in> A'", auto) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

446 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

447 
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

448 
show "b \<in> h ` A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
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diff
changeset

449 
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

450 
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

451 
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

452 
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

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

454 
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

455 
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

456 
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

457 
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

458 
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

459 
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

460 
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

461 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

462 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

463 
(* *) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

464 
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
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changeset

465 
qed 
14760  466 

467 
subsection {*Other Consequences of Hilbert's Epsilon*} 

468 

469 
text {*Hilbert's Epsilon and the @{term split} Operator*} 

470 

471 
text{*Looping simprule*} 

472 
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))" 

26347  473 
by simp 
14760  474 

475 
lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))" 

26347  476 
by (simp add: split_def) 
14760  477 

478 
lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)" 

26347  479 
by blast 
14760  480 

481 

482 
text{*A relation is wellfounded iff it has no infinite descending chain*} 

483 
lemma wf_iff_no_infinite_down_chain: 

484 
"wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))" 

485 
apply (simp only: wf_eq_minimal) 

486 
apply (rule iffI) 

487 
apply (rule notI) 

488 
apply (erule exE) 

489 
apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast) 

490 
apply (erule contrapos_np, simp, clarify) 

491 
apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q") 

492 
apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI) 

493 
apply (rule allI, simp) 

494 
apply (rule someI2_ex, blast, blast) 

495 
apply (rule allI) 

496 
apply (induct_tac "n", simp_all) 

497 
apply (rule someI2_ex, blast+) 

498 
done 

499 

27760  500 
lemma wf_no_infinite_down_chainE: 
501 
assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r" 

502 
using `wf r` wf_iff_no_infinite_down_chain[of r] by blast 

503 

504 

14760  505 
text{*A dynamicallyscoped fact for TFL *} 
12298  506 
lemma tfl_some: "\<forall>P x. P x > P (Eps P)" 
507 
by (blast intro: someI) 

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

508 

12298  509 

510 
subsection {* Least value operator *} 

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

511 

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

512 
definition 
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset

513 
LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where 
14760  514 
"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

515 

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

516 
syntax 
12298  517 
"_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

518 
translations 
35115  519 
"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

520 

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

521 
lemma LeastMI2: 
12298  522 
"P x ==> (!!y. P y ==> m x <= m y) 
523 
==> (!!x. P x ==> \<forall>y. P y > m x \<le> m y ==> Q x) 

524 
==> Q (LeastM m P)" 

14760  525 
apply (simp add: LeastM_def) 
14208  526 
apply (rule someI2_ex, blast, blast) 
12298  527 
done 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

528 

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

529 
lemma LeastM_equality: 
12298  530 
"P k ==> (!!x. P x ==> m k <= m x) 
531 
==> m (LEAST x WRT m. P x) = (m k::'a::order)" 

14208  532 
apply (rule LeastMI2, assumption, blast) 
12298  533 
apply (blast intro!: order_antisym) 
534 
done 

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

535 

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

536 
lemma wf_linord_ex_has_least: 
14760  537 
"wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k 
538 
==> \<exists>x. P x & (!y. P y > (m x,m y):r^*)" 

12298  539 
apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]]) 
14208  540 
apply (drule_tac x = "m`Collect P" in spec, force) 
12298  541 
done 
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset

542 

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

543 
lemma ex_has_least_nat: 
14760  544 
"P k ==> \<exists>x. P x & (\<forall>y. P y > m x <= (m y::nat))" 
12298  545 
apply (simp only: pred_nat_trancl_eq_le [symmetric]) 
546 
apply (rule wf_pred_nat [THEN wf_linord_ex_has_least]) 

16796  547 
apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption) 
12298  548 
done 
11454
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset

549 

12298  550 
lemma LeastM_nat_lemma: 
14760  551 
"P k ==> P (LeastM m P) & (\<forall>y. P y > m (LeastM m P) <= (m y::nat))" 
552 
apply (simp add: LeastM_def) 

12298  553 
apply (rule someI_ex) 
554 
apply (erule ex_has_least_nat) 

555 
done 

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

556 

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

557 
lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard] 
7514e5e21cb8
Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents:
11451
diff
changeset

558 

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

559 
lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)" 
14208  560 
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

561 

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

562 

12298  563 
subsection {* Greatest value operator *} 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

564 

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

565 
definition 
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset

566 
GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where 
14760  567 
"GreatestM m P == SOME x. P x & (\<forall>y. P y > m y <= m x)" 
12298  568 

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

569 
definition 
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35216
diff
changeset

570 
Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where 
12298  571 
"Greatest == GreatestM (%x. x)" 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

572 

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

573 
syntax 
35115  574 
"_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a" 
12298  575 
("GREATEST _ WRT _. _" [0, 4, 10] 10) 
11451
8abfb4f7bd02
partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff
changeset

576 
translations 
35115  577 
"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

578 

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

579 
lemma GreatestMI2: 
12298  580 
"P x ==> (!!y. P y ==> m y <= m x) 
581 
==> (!!x. P x ==> \<forall>y. P y > m y \<le> m x ==> Q x) 

582 
==> Q (GreatestM m P)" 

14760  583 
apply (simp add: GreatestM_def) 
14208  584 
apply (rule someI2_ex, blast, blast) 
12298  585 
done 
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586 

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lemma GreatestM_equality: 
12298  588 
"P k ==> (!!x. P x ==> m x <= m k) 
589 
==> m (GREATEST x WRT m. P x) = (m k::'a::order)" 

14208  590 
apply (rule_tac m = m in GreatestMI2, assumption, blast) 
12298  591 
apply (blast intro!: order_antisym) 
592 
done 

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593 

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lemma Greatest_equality: 
12298  595 
"P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k" 
14760  596 
apply (simp add: Greatest_def) 
14208  597 
apply (erule GreatestM_equality, blast) 
12298  598 
done 
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599 

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lemma ex_has_greatest_nat_lemma: 
14760  601 
"P k ==> \<forall>x. P x > (\<exists>y. P y & ~ ((m y::nat) <= m x)) 
602 
==> \<exists>y. P y & ~ (m y < m k + n)" 

15251  603 
apply (induct n, force) 
12298  604 
apply (force simp add: le_Suc_eq) 
605 
done 

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606 

12298  607 
lemma ex_has_greatest_nat: 
14760  608 
"P k ==> \<forall>y. P y > m y < b 
609 
==> \<exists>x. P x & (\<forall>y. P y > (m y::nat) <= m x)" 

12298  610 
apply (rule ccontr) 
611 
apply (cut_tac P = P and n = "b  m k" in ex_has_greatest_nat_lemma) 

14208  612 
apply (subgoal_tac [3] "m k <= b", auto) 
12298  613 
done 
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614 

12298  615 
lemma GreatestM_nat_lemma: 
14760  616 
"P k ==> \<forall>y. P y > m y < b 
617 
==> P (GreatestM m P) & (\<forall>y. P y > (m y::nat) <= m (GreatestM m P))" 

618 
apply (simp add: GreatestM_def) 

12298  619 
apply (rule someI_ex) 
14208  620 
apply (erule ex_has_greatest_nat, assumption) 
12298  621 
done 
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622 

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623 
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard] 
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12298  625 
lemma GreatestM_nat_le: 
14760  626 
"P x ==> \<forall>y. P y > m y < b 
12298  627 
==> (m x::nat) <= m (GreatestM m P)" 
21020  628 
apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P]) 
12298  629 
done 
630 

631 

632 
text {* \medskip Specialization to @{text GREATEST}. *} 

633 

14760  634 
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y > y < b ==> P (GREATEST x. P x)" 
635 
apply (simp add: Greatest_def) 

14208  636 
apply (rule GreatestM_natI, auto) 
12298  637 
done 
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638 

12298  639 
lemma Greatest_le: 
14760  640 
"P x ==> \<forall>y. P y > y < b ==> (x::nat) <= (GREATEST x. P x)" 
641 
apply (simp add: Greatest_def) 

14208  642 
apply (rule GreatestM_nat_le, auto) 
12298  643 
done 
644 

645 

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subsection {* Specification package  Hilbertized version *} 
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647 

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lemma exE_some: "[ Ex P ; c == Eps P ] ==> P c" 
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by (simp only: someI_ex) 
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650 

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use "Tools/choice_specification.ML" 
14115  652 

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653 
end 