author  haftmann 
Tue, 23 Jun 2009 11:31:28 +0200  
changeset 31769  d5f39775edd2 
parent 31731  7ffc1a901eea 
child 31770  ba52fcfaec28 
permissions  rwrr 
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(* Title: HOL/Library/FuncSet.thy 
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Author: Florian Kammueller and Lawrence C Paulson 

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*) 

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header {* Pi and Function Sets *} 
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theory FuncSet 
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Main is (Complex_Main) base entry point in library theories
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imports Hilbert_Choice Main 
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begin 
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definition 
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Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where 
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"Pi A B = {f. \<forall>x. x \<in> A > f x \<in> B x}" 
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definition 
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extensional :: "'a set => ('a => 'b) set" where 
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"extensional A = {f. \<forall>x. x~:A > f x = undefined}" 
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definition 
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"restrict" :: "['a => 'b, 'a set] => ('a => 'b)" where 
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"restrict f A = (%x. if x \<in> A then f x else undefined)" 
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abbreviation 
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funcset :: "['a set, 'b set] => ('a => 'b) set" 
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(infixr ">" 60) where 
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"A > B == Pi A (%_. B)" 
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notation (xsymbols) 
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tuned concrete syntax  abbreviation/const_syntax;
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funcset (infixr "\<rightarrow>" 60) 
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syntax 
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"_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3PI _:_./ _)" 10) 
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"_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)" ("(3%_:_./ _)" [0,0,3] 3) 

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syntax (xsymbols) 

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"_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi> _\<in>_./ _)" 10) 
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"_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)" ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3) 

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syntax (HTML output) 
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"_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi> _\<in>_./ _)" 10) 
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"_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)" ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3) 

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translations 
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"PI x:A. B" == "CONST Pi A (%x. B)" 
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"%x:A. f" == "CONST restrict (%x. f) A" 

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definition 
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"compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)" where 
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"compose A g f = (\<lambda>x\<in>A. g (f x))" 
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subsection{*Basic Properties of @{term Pi}*} 

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lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B" 
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by (simp add: Pi_def) 
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lemma Pi_I'[simp]: "(!!x. x : A > f x : B x) ==> f : Pi A B" 
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by(simp add:Pi_def) 

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lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A > B" 
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by (simp add: Pi_def) 
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lemma Pi_mem: "[f: Pi A B; x \<in> A] ==> f x \<in> B x" 

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by (simp add: Pi_def) 
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lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A" 
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by (auto intro: Pi_I) 

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lemma funcset_mem: "[f \<in> A > B; x \<in> A] ==> f x \<in> B" 
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by (simp add: Pi_def) 
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lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B" 
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by (auto simp add: Pi_def) 
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lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})" 
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apply (simp add: Pi_def, auto) 
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txt{*Converse direction requires Axiom of Choice to exhibit a function 
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picking an element from each nonempty @{term "B x"}*} 

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apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto) 
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apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto) 
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done 
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lemma Pi_empty [simp]: "Pi {} B = UNIV" 
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by (simp add: Pi_def) 
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lemma Pi_UNIV [simp]: "A > UNIV = UNIV" 

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by (simp add: Pi_def) 
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(* 
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lemma funcset_id [simp]: "(%x. x): A > A" 

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by (simp add: Pi_def) 

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*) 

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text{*Covariance of Pisets in their second argument*} 
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lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C" 

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by (simp add: Pi_def, blast) 
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text{*Contravariance of Pisets in their first argument*} 

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lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B" 

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by (simp add: Pi_def, blast) 
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subsection{*Composition With a Restricted Domain: @{term compose}*} 

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lemma funcset_compose: 
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"[ f \<in> A > B; g \<in> B > C ]==> compose A g f \<in> A > C" 

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by (simp add: Pi_def compose_def restrict_def) 

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lemma compose_assoc: 

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"[ f \<in> A > B; g \<in> B > C; h \<in> C > D ] 
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==> compose A h (compose A g f) = compose A (compose B h g) f" 
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by (simp add: expand_fun_eq Pi_def compose_def restrict_def) 
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lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))" 

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by (simp add: compose_def restrict_def) 
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lemma surj_compose: "[ f ` A = B; g ` B = C ] ==> compose A g f ` A = C" 

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by (auto simp add: image_def compose_eq) 
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subsection{*Bounded Abstraction: @{term restrict}*} 

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lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A > B" 

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by (simp add: Pi_def restrict_def) 
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lemma restrictI: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B" 

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by (simp add: Pi_def restrict_def) 
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lemma restrict_apply [simp]: 

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"(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)" 
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by (simp add: restrict_def) 
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lemma restrict_ext: 
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"(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)" 
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by (simp add: expand_fun_eq Pi_def Pi_def restrict_def) 
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lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A" 
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by (simp add: inj_on_def restrict_def) 
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lemma Id_compose: 

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"[f \<in> A > B; f \<in> extensional A] ==> compose A (\<lambda>y\<in>B. y) f = f" 
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by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def) 

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lemma compose_Id: 

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"[g \<in> A > B; g \<in> extensional A] ==> compose A g (\<lambda>x\<in>A. x) = g" 
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by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def) 

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lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A" 
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by (auto simp add: restrict_def) 
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subsection{*Bijections Between Sets*} 
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text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of 
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the theorems belong here, or need at least @{term Hilbert_Choice}.*} 
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lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B" 

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by (auto simp add: bij_betw_def inj_on_Inv Pi_def) 
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lemma inj_on_compose: 
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"[ bij_betw f A B; inj_on g B ] ==> inj_on (compose A g f) A" 

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by (auto simp add: bij_betw_def inj_on_def compose_eq) 

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lemma bij_betw_compose: 
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"[ bij_betw f A B; bij_betw g B C ] ==> bij_betw (compose A g f) A C" 

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apply (simp add: bij_betw_def compose_eq inj_on_compose) 
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apply (auto simp add: compose_def image_def) 

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done 

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lemma bij_betw_restrict_eq [simp]: 
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"bij_betw (restrict f A) A B = bij_betw f A B" 

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by (simp add: bij_betw_def) 

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subsection{*Extensionality*} 

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lemma extensional_arb: "[f \<in> extensional A; x\<notin> A] ==> f x = undefined" 
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by (simp add: extensional_def) 
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lemma restrict_extensional [simp]: "restrict f A \<in> extensional A" 

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by (simp add: restrict_def extensional_def) 

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lemma compose_extensional [simp]: "compose A f g \<in> extensional A" 

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by (simp add: compose_def) 

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lemma extensionalityI: 

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"[ f \<in> extensional A; g \<in> extensional A; 

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!!x. x\<in>A ==> f x = g x ] ==> f = g" 

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by (force simp add: expand_fun_eq extensional_def) 

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lemma Inv_funcset: "f ` A = B ==> (\<lambda>x\<in>B. Inv A f x) : B > A" 

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by (unfold Inv_def) (fast intro: restrict_in_funcset someI2) 

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lemma compose_Inv_id: 

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"bij_betw f A B ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)" 

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apply (simp add: bij_betw_def compose_def) 

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apply (rule restrict_ext, auto) 

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apply (erule subst) 

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apply (simp add: Inv_f_f) 

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done 

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lemma compose_id_Inv: 

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"f ` A = B ==> compose B f (\<lambda>y\<in>B. Inv A f y) = (\<lambda>x\<in>B. x)" 

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apply (simp add: compose_def) 

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apply (rule restrict_ext) 

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apply (simp add: f_Inv_f) 

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done 

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subsection{*Cardinality*} 
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lemma card_inj: "[f \<in> A\<rightarrow>B; inj_on f A; finite B] ==> card(A) \<le> card(B)" 

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apply (rule card_inj_on_le) 
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apply (auto simp add: Pi_def) 

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done 

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lemma card_bij: 

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"[f \<in> A\<rightarrow>B; inj_on f A; 

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g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B] ==> card(A) = card(B)" 

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by (blast intro: card_inj order_antisym) 
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end 