author  haftmann 
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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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Plain, Main form meeting points in import hierarchy
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imports Nat Wellfounded Plain 
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uses ("Tools/meson.ML") ("Tools/choice_specification.ML") 
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begin 
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13 
subsection {* Hilbert's epsilon *} 

14 

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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) = atomic_abs_tr' abs 

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in Syntax.const @{syntax_const "_Eps"} $ x $ t end)] 

31 
*}  {* 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 

40 
subsection {*Hilbert's Epsilonoperator*} 

41 

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

43 
existential formula*} 

44 
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) 

47 
done 

48 

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

50 
occurrence of @{term P}.*} 

51 
lemma someI2: "[ P a; !!x. P x ==> Q x ] ==> Q (SOME x. P x)" 

52 
by (blast intro: someI) 

53 

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

55 
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) 

58 

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lemma some_equality [intro]: 

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"[ P a; !!x. P x ==> x=a ] ==> (SOME x. P x) = a" 

61 
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 
14760  65 

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lemma some_eq_ex: "P (SOME x. P x) = (\<exists>x. P x)" 

67 
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 

73 

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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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80 

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

82 

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text{*Used in @{text "Tools/meson.ML"}*} 

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lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)" 

85 
by (fast elim: someI) 

86 

87 
lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)" 

88 
by (fast elim: someI) 

89 

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subsection {*Function Inverse*} 

92 

33014  93 
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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33057  96 
lemma inv_into_into: "x : f ` A ==> inv_into A f x : A" 
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apply (simp add: inv_into_def) 

32988  98 
apply (fast intro: someI2) 
99 
done 

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32988  101 
lemma inv_id [simp]: "inv id = id" 
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by (simp add: inv_into_def id_def) 
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33057  104 
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 
109 

32988  110 
lemma inv_f_f: "inj f ==> inv f (f x) = x" 
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by simp 
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33057  113 
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) 
116 
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: ext inv_into_f_eq) 
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text{*But is it useful?*} 

130 
lemma inj_transfer: 

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assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)" 

132 
shows "P x" 

133 
proof  

134 
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 expand_fun_eq) 

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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" 
145 
by (simp add: inj_iff) 

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lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g" 

148 
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(fastsimp 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" 

33057  158 
by (simp add: f_inv_into_f surj_range) 
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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) 

14760  168 
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)" 

33057  177 
by (simp add: inj_on_inv_into surj_range) 
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lemma surj_iff: "(surj f) = (f o inv f = id)" 

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apply (simp add: o_def expand_fun_eq) 

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apply (blast intro: surjI surj_f_inv_f) 

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done 

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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) 

222 
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)}" 

234 
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]) 

237 
done 

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lemma bij_vimage_eq_inv_image: "bij f ==> f ` A = inv f ` A" 

240 
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]) 
14760  242 
done 
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lemma finite_fun_UNIVD1: 
245 
assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)" 

246 
and card: "card (UNIV :: 'b set) \<noteq> Suc 0" 

247 
shows "finite (UNIV :: 'a set)" 

248 
proof  

249 
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)" 

256 
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 

261 
ultimately show "finite (UNIV :: 'a set)" by simp 

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qed 

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

266 

267 
text {*Hilbert's Epsilon and the @{term split} Operator*} 

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text{*Looping simprule*} 

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

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by simp 
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273 
lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))" 

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

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

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lemma wf_iff_no_infinite_down_chain: 

282 
"wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))" 

283 
apply (simp only: wf_eq_minimal) 

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apply (rule iffI) 

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apply (rule notI) 

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apply (erule exE) 

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

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apply (erule contrapos_np, simp, clarify) 

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

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

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apply (rule allI, simp) 

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apply (rule someI2_ex, blast, blast) 

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apply (rule allI) 

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apply (induct_tac "n", simp_all) 

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apply (rule someI2_ex, blast+) 

296 
done 

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

300 
using `wf r` wf_iff_no_infinite_down_chain[of r] by blast 

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14760  303 
text{*A dynamicallyscoped fact for TFL *} 
12298  304 
lemma tfl_some: "\<forall>P x. P x > P (Eps P)" 
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by (blast intro: someI) 

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12298  307 

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subsection {* Least value operator *} 

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definition 
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LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where 
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"LeastM m P == SOME x. P x & (\<forall>y. P y > m x <= m y)" 
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syntax 
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"_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a" ("LEAST _ WRT _. _" [0, 4, 10] 10) 
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translations 
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"LEAST x WRT m. P" == "CONST LeastM m (%x. P)" 
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lemma LeastMI2: 
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"P x ==> (!!y. P y ==> m x <= m y) 
321 
==> (!!x. P x ==> \<forall>y. P y > m x \<le> m y ==> Q x) 

322 
==> Q (LeastM m P)" 

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apply (simp add: LeastM_def) 
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apply (rule someI2_ex, blast, blast) 
12298  325 
done 
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lemma LeastM_equality: 
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"P k ==> (!!x. P x ==> m k <= m x) 
329 
==> m (LEAST x WRT m. P x) = (m k::'a::order)" 

14208  330 
apply (rule LeastMI2, assumption, blast) 
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apply (blast intro!: order_antisym) 
332 
done 

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lemma wf_linord_ex_has_least: 
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"wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k 
336 
==> \<exists>x. P x & (!y. P y > (m x,m y):r^*)" 

12298  337 
apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]]) 
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apply (drule_tac x = "m`Collect P" in spec, force) 
12298  339 
done 
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lemma ex_has_least_nat: 
14760  342 
"P k ==> \<exists>x. P x & (\<forall>y. P y > m x <= (m y::nat))" 
12298  343 
apply (simp only: pred_nat_trancl_eq_le [symmetric]) 
344 
apply (rule wf_pred_nat [THEN wf_linord_ex_has_least]) 

16796  345 
apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption) 
12298  346 
done 
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12298  348 
lemma LeastM_nat_lemma: 
14760  349 
"P k ==> P (LeastM m P) & (\<forall>y. P y > m (LeastM m P) <= (m y::nat))" 
350 
apply (simp add: LeastM_def) 

12298  351 
apply (rule someI_ex) 
352 
apply (erule ex_has_least_nat) 

353 
done 

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lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard] 
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lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)" 
14208  358 
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption) 
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359 

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12298  361 
subsection {* Greatest value operator *} 
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362 

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definition 
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GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where 
14760  365 
"GreatestM m P == SOME x. P x & (\<forall>y. P y > m y <= m x)" 
12298  366 

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definition 
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Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where 
12298  369 
"Greatest == GreatestM (%x. x)" 
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syntax 
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"_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a" 
12298  373 
("GREATEST _ WRT _. _" [0, 4, 10] 10) 
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translations 
35115  375 
"GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)" 
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376 

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lemma GreatestMI2: 
12298  378 
"P x ==> (!!y. P y ==> m y <= m x) 
379 
==> (!!x. P x ==> \<forall>y. P y > m y \<le> m x ==> Q x) 

380 
==> Q (GreatestM m P)" 

14760  381 
apply (simp add: GreatestM_def) 
14208  382 
apply (rule someI2_ex, blast, blast) 
12298  383 
done 
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384 

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lemma GreatestM_equality: 
12298  386 
"P k ==> (!!x. P x ==> m x <= m k) 
387 
==> m (GREATEST x WRT m. P x) = (m k::'a::order)" 

14208  388 
apply (rule_tac m = m in GreatestMI2, assumption, blast) 
12298  389 
apply (blast intro!: order_antisym) 
390 
done 

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391 

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lemma Greatest_equality: 
12298  393 
"P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k" 
14760  394 
apply (simp add: Greatest_def) 
14208  395 
apply (erule GreatestM_equality, blast) 
12298  396 
done 
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397 

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lemma ex_has_greatest_nat_lemma: 
14760  399 
"P k ==> \<forall>x. P x > (\<exists>y. P y & ~ ((m y::nat) <= m x)) 
400 
==> \<exists>y. P y & ~ (m y < m k + n)" 

15251  401 
apply (induct n, force) 
12298  402 
apply (force simp add: le_Suc_eq) 
403 
done 

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404 

12298  405 
lemma ex_has_greatest_nat: 
14760  406 
"P k ==> \<forall>y. P y > m y < b 
407 
==> \<exists>x. P x & (\<forall>y. P y > (m y::nat) <= m x)" 

12298  408 
apply (rule ccontr) 
409 
apply (cut_tac P = P and n = "b  m k" in ex_has_greatest_nat_lemma) 

14208  410 
apply (subgoal_tac [3] "m k <= b", auto) 
12298  411 
done 
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412 

12298  413 
lemma GreatestM_nat_lemma: 
14760  414 
"P k ==> \<forall>y. P y > m y < b 
415 
==> P (GreatestM m P) & (\<forall>y. P y > (m y::nat) <= m (GreatestM m P))" 

416 
apply (simp add: GreatestM_def) 

12298  417 
apply (rule someI_ex) 
14208  418 
apply (erule ex_has_greatest_nat, assumption) 
12298  419 
done 
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lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard] 
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12298  423 
lemma GreatestM_nat_le: 
14760  424 
"P x ==> \<forall>y. P y > m y < b 
12298  425 
==> (m x::nat) <= m (GreatestM m P)" 
21020  426 
apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P]) 
12298  427 
done 
428 

429 

430 
text {* \medskip Specialization to @{text GREATEST}. *} 

431 

14760  432 
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y > y < b ==> P (GREATEST x. P x)" 
433 
apply (simp add: Greatest_def) 

14208  434 
apply (rule GreatestM_natI, auto) 
12298  435 
done 
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12298  437 
lemma Greatest_le: 
14760  438 
"P x ==> \<forall>y. P y > y < b ==> (x::nat) <= (GREATEST x. P x)" 
439 
apply (simp add: Greatest_def) 

14208  440 
apply (rule GreatestM_nat_le, auto) 
12298  441 
done 
442 

443 

444 
subsection {* The Meson proof procedure *} 

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12298  446 
subsubsection {* Negation Normal Form *} 
447 

448 
text {* de Morgan laws *} 

449 

450 
lemma meson_not_conjD: "~(P&Q) ==> ~P  ~Q" 

451 
and meson_not_disjD: "~(PQ) ==> ~P & ~Q" 

452 
and meson_not_notD: "~~P ==> P" 

14760  453 
and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)" 
454 
and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)" 

12298  455 
by fast+ 
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456 

12298  457 
text {* Removal of @{text ">"} and @{text "<>"} (positive and 
458 
negative occurrences) *} 

459 

460 
lemma meson_imp_to_disjD: "P>Q ==> ~P  Q" 

461 
and meson_not_impD: "~(P>Q) ==> P & ~Q" 

462 
and meson_iff_to_disjD: "P=Q ==> (~P  Q) & (~Q  P)" 

463 
and meson_not_iffD: "~(P=Q) ==> (P  Q) & (~P  ~Q)" 

464 
 {* Much more efficient than @{prop "(P & ~Q)  (Q & ~P)"} for computing CNF *} 

18389  465 
and meson_not_refl_disj_D: "x ~= x  P ==> P" 
12298  466 
by fast+ 
467 

468 

469 
subsubsection {* Pulling out the existential quantifiers *} 

470 

471 
text {* Conjunction *} 

472 

14760  473 
lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q" 
474 
and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)" 

12298  475 
by fast+ 
476 

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12298  478 
text {* Disjunction *} 
479 

14760  480 
lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x))  (\<exists>x. Q(x)) ==> \<exists>x. P(x)  Q(x)" 
12298  481 
 {* DO NOT USE with forallSkolemization: makes fewer schematic variables!! *} 
482 
 {* With exSkolemization, makes fewer Skolem constants *} 

14760  483 
and meson_disj_exD1: "!!P Q. (\<exists>x. P(x))  Q ==> \<exists>x. P(x)  Q" 
484 
and meson_disj_exD2: "!!P Q. P  (\<exists>x. Q(x)) ==> \<exists>x. P  Q(x)" 

12298  485 
by fast+ 
486 

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12298  488 
subsubsection {* Generating clauses for the Meson Proof Procedure *} 
489 

490 
text {* Disjunctions *} 

491 

492 
lemma meson_disj_assoc: "(PQ)R ==> P(QR)" 

493 
and meson_disj_comm: "PQ ==> QP" 

494 
and meson_disj_FalseD1: "FalseP ==> P" 

495 
and meson_disj_FalseD2: "PFalse ==> P" 

496 
by fast+ 

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14760  498 

499 
subsection{*Lemmas for Meson, the Model Elimination Procedure*} 

500 

501 
text{* Generation of contrapositives *} 

502 

503 
text{*Inserts negated disjunct after removing the negation; P is a literal. 

504 
Model elimination requires assuming the negation of every attempted subgoal, 

505 
hence the negated disjuncts.*} 

506 
lemma make_neg_rule: "~PQ ==> ((~P==>P) ==> Q)" 

507 
by blast 

508 

509 
text{*Version for Plaisted's "Postive refinement" of the Meson procedure*} 

510 
lemma make_refined_neg_rule: "~PQ ==> (P ==> Q)" 

511 
by blast 

512 

513 
text{*@{term P} should be a literal*} 

514 
lemma make_pos_rule: "PQ ==> ((P==>~P) ==> Q)" 

515 
by blast 

516 

517 
text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't 

518 
insert new assumptions, for ordinary resolution.*} 

519 

520 
lemmas make_neg_rule' = make_refined_neg_rule 

521 

522 
lemma make_pos_rule': "[PQ; ~P] ==> Q" 

523 
by blast 

524 

525 
text{* Generation of a goal clause  put away the final literal *} 

526 

527 
lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)" 

528 
by blast 

529 

530 
lemma make_pos_goal: "P ==> ((P==>~P) ==> False)" 

531 
by blast 

532 

533 

534 
subsubsection{* Lemmas for Forward Proof*} 

535 

536 
text{*There is a similarity to congruence rules*} 

537 

538 
(*NOTE: could handle conjunctions (faster?) by 

539 
nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *) 

540 
lemma conj_forward: "[ P'&Q'; P' ==> P; Q' ==> Q ] ==> P&Q" 

541 
by blast 

542 

543 
lemma disj_forward: "[ P'Q'; P' ==> P; Q' ==> Q ] ==> PQ" 

544 
by blast 

545 

546 
(*Version of @{text disj_forward} for removal of duplicate literals*) 

547 
lemma disj_forward2: 

548 
"[ P'Q'; P' ==> P; [ Q'; P==>False ] ==> Q ] ==> PQ" 

549 
apply blast 

550 
done 

551 

552 
lemma all_forward: "[ \<forall>x. P'(x); !!x. P'(x) ==> P(x) ] ==> \<forall>x. P(x)" 

553 
by blast 

554 

555 
lemma ex_forward: "[ \<exists>x. P'(x); !!x. P'(x) ==> P(x) ] ==> \<exists>x. P(x)" 

556 
by blast 

557 

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subsection {* Meson package *} 
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560 

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use "Tools/meson.ML" 
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562 

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setup Meson.setup 
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564 

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565 

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566 
subsection {* Specification package  Hilbertized version *} 
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567 

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lemma exE_some: "[ Ex P ; c == Eps P ] ==> P c" 
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569 
by (simp only: someI_ex) 
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570 

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use "Tools/choice_specification.ML" 
14115  572 

31454  573 

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574 
end 