author  wenzelm 
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permissions  rwrr 
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(* Title: HOL/Library/FuncSet.thy 
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Author: Florian Kammueller and Lawrence C Paulson, Lukas Bulwahn 
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*) 
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section \<open>Pi and Function Sets\<close> 
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theory FuncSet 
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imports Hilbert_Choice Main 
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begin 
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definition Pi :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set" 
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where "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}" 

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definition extensional :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) set" 
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where "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}" 

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definition "restrict" :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" 
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where "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)" 

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abbreviation funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr ">" 60) 
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where "A > B \<equiv> Pi A (\<lambda>_. B)" 

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notation (xsymbols) 
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funcset (infixr "\<rightarrow>" 60) 
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syntax 
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"_Pi" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" ("(3PI _:_./ _)" 10) 
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"_lam" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> 'b)" ("(3%_:_./ _)" [0,0,3] 3) 

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

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translations 
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"\<Pi> x\<in>A. B" \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)" 
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"\<lambda>x\<in>A. f" \<rightleftharpoons> "CONST restrict (\<lambda>x. f) A" 

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definition "compose" :: "'a set \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c)" 
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where "compose A g f = (\<lambda>x\<in>A. g (f x))" 

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subsection \<open>Basic Properties of @{term Pi}\<close> 
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lemma Pi_I[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B" 
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by (simp add: Pi_def) 
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lemma Pi_I'[simp]: "(\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B" 
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by (simp add:Pi_def) 

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lemma funcsetI: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f \<in> A \<rightarrow> B" 
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by (simp add: Pi_def) 
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lemma Pi_mem: "f \<in> Pi A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x" 
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by (simp add: Pi_def) 
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lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)" 
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unfolding Pi_def by auto 

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lemma PiE [elim]: "f \<in> Pi A B \<Longrightarrow> (f x \<in> B x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q" 
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by (auto simp: Pi_def) 

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lemma Pi_cong: "(\<And>w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi A B \<longleftrightarrow> g \<in> Pi A B" 
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by (auto simp: Pi_def) 
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lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A" 
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by auto 
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lemma funcset_mem: "f \<in> A \<rightarrow> B \<Longrightarrow> x \<in> A \<Longrightarrow> 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 \<Longrightarrow> f ` A \<subseteq> B" 
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by auto 
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lemma image_subset_iff_funcset: "F ` A \<subseteq> B \<longleftrightarrow> F \<in> A \<rightarrow> B" 

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by auto 

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lemma Pi_eq_empty[simp]: "(\<Pi> x \<in> A. B x) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})" 
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apply (simp add: Pi_def) 

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apply auto 

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txt \<open>Converse direction requires Axiom of Choice to exhibit a function 

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picking an element from each nonempty @{term "B x"}\<close> 

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apply (drule_tac x = "\<lambda>u. SOME y. y \<in> B u" in spec) 

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apply auto 

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apply (cut_tac P = "\<lambda>y. y \<in> B x" in some_eq_ex) 

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apply 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_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)" 
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by auto 
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lemma Pi_UN: 
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fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set" 
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assumes "finite I" 
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and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i" 

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shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)" 
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proof (intro set_eqI iffI) 
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fix f 
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assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)" 

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then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" 

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by auto 

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from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" 

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by auto 

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obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k" 
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using \<open>finite I\<close> finite_nat_set_iff_bounded_le[of "n`I"] by auto 
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have "f \<in> Pi I (A k)" 
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proof (intro Pi_I) 
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fix i 
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assume "i \<in> I" 

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from mono[OF this, of "n i" k] k[OF this] n[OF this] 
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show "f i \<in> A k i" by auto 
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qed 
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then show "f \<in> (\<Union>n. Pi I (A n))" 
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by auto 

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qed auto 
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lemma Pi_UNIV [simp]: "A \<rightarrow> UNIV = UNIV" 
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by (simp add: Pi_def) 

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text \<open>Covariance of Pisets in their second argument\<close> 
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lemma Pi_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi A B \<subseteq> Pi A C" 

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by auto 

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text \<open>Contravariance of Pisets in their first argument\<close> 
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lemma Pi_anti_mono: "A' \<subseteq> A \<Longrightarrow> Pi A B \<subseteq> Pi A' B" 

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by auto 

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lemma prod_final: 
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assumes 1: "fst \<circ> f \<in> Pi A B" 
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and 2: "snd \<circ> f \<in> Pi A C" 

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shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)" 
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proof (rule Pi_I) 
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fix z 
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assume z: "z \<in> A" 
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have "f z = (fst (f z), snd (f z))" 

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by simp 
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also have "\<dots> \<in> B z \<times> C z" 
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by (metis SigmaI PiE o_apply 1 2 z) 

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finally show "f z \<in> B z \<times> C z" . 
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qed 
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lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X" 
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by (auto simp: Pi_def) 
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lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i" 
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by (auto simp: Pi_def) 
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lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B" 
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by (auto simp: Pi_def) 
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lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" 
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by (auto simp: Pi_def) 
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lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I  {i}) B \<and> f i \<in> A" 
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apply auto 
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apply (drule_tac x=x in Pi_mem) 
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apply (simp_all split: split_if_asm) 
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apply (drule_tac x=i in Pi_mem) 
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apply (auto dest!: Pi_mem) 
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done 
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subsection \<open>Composition With a Restricted Domain: @{term compose}\<close> 

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lemma funcset_compose: "f \<in> A \<rightarrow> B \<Longrightarrow> g \<in> B \<rightarrow> C \<Longrightarrow> compose A g f \<in> A \<rightarrow> 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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assumes "f \<in> A \<rightarrow> B" 
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and "g \<in> B \<rightarrow> C" 

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and "h \<in> C \<rightarrow> D" 

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shows "compose A h (compose A g f) = compose A (compose B h g) f" 

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using assms by (simp add: fun_eq_iff Pi_def compose_def restrict_def) 

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lemma compose_eq: "x \<in> A \<Longrightarrow> 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 \<Longrightarrow> g ` B = C \<Longrightarrow> compose A g f ` A = C" 
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by (auto simp add: image_def compose_eq) 
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subsection \<open>Bounded Abstraction: @{term restrict}\<close> 
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lemma restrict_cong: "I = J \<Longrightarrow> (\<And>i. i \<in> J =simp=> f i = g i) \<Longrightarrow> restrict f I = restrict g J" 
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by (auto simp: restrict_def fun_eq_iff simp_implies_def) 
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lemma restrict_in_funcset: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> (\<lambda>x\<in>A. f x) \<in> A \<rightarrow> B" 
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by (simp add: Pi_def restrict_def) 
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lemma restrictI[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> (\<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]: "(\<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_apply': "x \<in> A \<Longrightarrow> (\<lambda>y\<in>A. f y) x = f x" 
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by simp 

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lemma restrict_ext: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)" 
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by (simp add: fun_eq_iff Pi_def restrict_def) 
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lemma restrict_UNIV: "restrict f UNIV = f" 
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by (simp add: 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: "f \<in> A \<rightarrow> B \<Longrightarrow> f \<in> extensional A \<Longrightarrow> compose A (\<lambda>y\<in>B. y) f = f" 
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by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def) 
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lemma compose_Id: "g \<in> A \<rightarrow> B \<Longrightarrow> g \<in> extensional A \<Longrightarrow> compose A g (\<lambda>x\<in>A. x) = g" 
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by (auto simp add: fun_eq_iff 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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lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)" 
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unfolding restrict_def by (simp add: fun_eq_iff) 
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lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I" 
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by (auto simp: restrict_def) 
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lemma restrict_upd[simp]: "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)" 
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by (auto simp: fun_eq_iff) 
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lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A" 
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by (auto simp: restrict_def Pi_def) 
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14745  229 

58783  230 
subsection \<open>Bijections Between Sets\<close> 
14762  231 

58783  232 
text \<open>The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of 
233 
the theorems belong here, or need at least @{term Hilbert_Choice}.\<close> 

14762  234 

39595  235 
lemma bij_betwI: 
58783  236 
assumes "f \<in> A \<rightarrow> B" 
237 
and "g \<in> B \<rightarrow> A" 

238 
and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x" 

239 
and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y" 

240 
shows "bij_betw f A B" 

241 
unfolding bij_betw_def 

39595  242 
proof 
58783  243 
show "inj_on f A" 
244 
by (metis g_f inj_on_def) 

245 
have "f ` A \<subseteq> B" 

246 
using \<open>f \<in> A \<rightarrow> B\<close> by auto 

39595  247 
moreover 
58783  248 
have "B \<subseteq> f ` A" 
249 
by auto (metis Pi_mem \<open>g \<in> B \<rightarrow> A\<close> f_g image_iff) 

250 
ultimately show "f ` A = B" 

251 
by blast 

39595  252 
qed 
253 

14762  254 
lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B" 
58783  255 
by (auto simp add: bij_betw_def) 
14762  256 

58783  257 
lemma inj_on_compose: "bij_betw f A B \<Longrightarrow> inj_on g B \<Longrightarrow> inj_on (compose A g f) A" 
258 
by (auto simp add: bij_betw_def inj_on_def compose_eq) 

14853  259 

58783  260 
lemma bij_betw_compose: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (compose A g f) A C" 
261 
apply (simp add: bij_betw_def compose_eq inj_on_compose) 

262 
apply (auto simp add: compose_def image_def) 

263 
done 

14762  264 

58783  265 
lemma bij_betw_restrict_eq [simp]: "bij_betw (restrict f A) A B = bij_betw f A B" 
266 
by (simp add: bij_betw_def) 

14853  267 

268 

58783  269 
subsection \<open>Extensionality\<close> 
14853  270 

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lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}" 
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unfolding extensional_def by auto 
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58783  274 
lemma extensional_arb: "f \<in> extensional A \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = undefined" 
275 
by (simp add: extensional_def) 

14853  276 

277 
lemma restrict_extensional [simp]: "restrict f A \<in> extensional A" 

58783  278 
by (simp add: restrict_def extensional_def) 
14853  279 

280 
lemma compose_extensional [simp]: "compose A f g \<in> extensional A" 

58783  281 
by (simp add: compose_def) 
14853  282 

283 
lemma extensionalityI: 

58783  284 
assumes "f \<in> extensional A" 
285 
and "g \<in> extensional A" 

286 
and "\<And>x. x \<in> A \<Longrightarrow> f x = g x" 

287 
shows "f = g" 

288 
using assms by (force simp add: fun_eq_iff extensional_def) 

14853  289 

39595  290 
lemma extensional_restrict: "f \<in> extensional A \<Longrightarrow> restrict f A = f" 
58783  291 
by (rule extensionalityI[OF restrict_extensional]) auto 
39595  292 

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lemma extensional_subset: "f \<in> extensional A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f \<in> extensional B" 
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unfolding extensional_def by auto 
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58783  296 
lemma inv_into_funcset: "f ` A = B \<Longrightarrow> (\<lambda>x\<in>B. inv_into A f x) \<in> B \<rightarrow> A" 
297 
by (unfold inv_into_def) (fast intro: someI2) 

14853  298 

58783  299 
lemma compose_inv_into_id: "bij_betw f A B \<Longrightarrow> compose A (\<lambda>y\<in>B. inv_into A f y) f = (\<lambda>x\<in>A. x)" 
300 
apply (simp add: bij_betw_def compose_def) 

301 
apply (rule restrict_ext, auto) 

302 
done 

14853  303 

58783  304 
lemma compose_id_inv_into: "f ` A = B \<Longrightarrow> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)" 
305 
apply (simp add: compose_def) 

306 
apply (rule restrict_ext) 

307 
apply (simp add: f_inv_into_f) 

308 
done 

14853  309 

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lemma extensional_insert[intro, simp]: 
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assumes "a \<in> extensional (insert i I)" 
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shows "a(i := b) \<in> extensional (insert i I)" 
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using assms unfolding extensional_def by auto 
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58783  315 
lemma extensional_Int[simp]: "extensional I \<inter> extensional I' = extensional (I \<inter> I')" 
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unfolding extensional_def by auto 
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317 

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lemma extensional_UNIV[simp]: "extensional UNIV = UNIV" 
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by (auto simp: extensional_def) 
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320 

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lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B" 
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unfolding restrict_def extensional_def by auto 
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323 

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lemma extensional_insert_undefined[intro, simp]: 
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"a \<in> extensional (insert i I) \<Longrightarrow> a(i := undefined) \<in> extensional I" 
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unfolding extensional_def by auto 
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327 

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lemma extensional_insert_cancel[intro, simp]: 
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"a \<in> extensional I \<Longrightarrow> a \<in> extensional (insert i I)" 
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unfolding extensional_def by auto 
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331 

14762  332 

58783  333 
subsection \<open>Cardinality\<close> 
14745  334 

58783  335 
lemma card_inj: "f \<in> A \<rightarrow> B \<Longrightarrow> inj_on f A \<Longrightarrow> finite B \<Longrightarrow> card A \<le> card B" 
336 
by (rule card_inj_on_le) auto 

14745  337 

338 
lemma card_bij: 

58783  339 
assumes "f \<in> A \<rightarrow> B" "inj_on f A" 
340 
and "g \<in> B \<rightarrow> A" "inj_on g B" 

341 
and "finite A" "finite B" 

342 
shows "card A = card B" 

343 
using assms by (blast intro: card_inj order_antisym) 

14745  344 

58783  345 

346 
subsection \<open>Extensional Function Spaces\<close> 

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58783  348 
definition PiE :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set" 
349 
where "PiE S T = Pi S T \<inter> extensional S" 

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350 

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abbreviation "Pi\<^sub>E A B \<equiv> PiE A B" 
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58783  353 
syntax 
354 
"_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" ("(3PIE _:_./ _)" 10) 

355 
syntax (xsymbols) 

356 
"_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10) 

357 
translations "\<Pi>\<^sub>E x\<in>A. B" \<rightleftharpoons> "CONST Pi\<^sub>E A (\<lambda>x. B)" 

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358 

58783  359 
abbreviation extensional_funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr ">\<^sub>E" 60) 
360 
where "A >\<^sub>E B \<equiv> (\<Pi>\<^sub>E i\<in>A. B)" 

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361 

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notation (xsymbols) 
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extensional_funcset (infixr "\<rightarrow>\<^sub>E" 60) 
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364 

58783  365 
lemma extensional_funcset_def: "extensional_funcset S T = (S \<rightarrow> T) \<inter> extensional S" 
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366 
by (simp add: PiE_def) 
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367 

58783  368 
lemma PiE_empty_domain[simp]: "PiE {} T = {\<lambda>x. undefined}" 
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369 
unfolding PiE_def by simp 
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370 

54417  371 
lemma PiE_UNIV_domain: "PiE UNIV T = Pi UNIV T" 
372 
unfolding PiE_def by simp 

373 

58783  374 
lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (\<Pi>\<^sub>E i\<in>I. F i) = {}" 
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375 
unfolding PiE_def by auto 
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376 

58783  377 
lemma PiE_eq_empty_iff: "Pi\<^sub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})" 
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378 
proof 
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379 
assume "Pi\<^sub>E I F = {}" 
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380 
show "\<exists>i\<in>I. F i = {}" 
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381 
proof (rule ccontr) 
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382 
assume "\<not> ?thesis" 
58783  383 
then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" 
384 
by auto 

53381  385 
from choice[OF this] 
386 
obtain f where " \<forall>x. (x \<in> I \<longrightarrow> f x \<in> F x) \<and> (x \<notin> I \<longrightarrow> f x = undefined)" .. 

58783  387 
then have "f \<in> Pi\<^sub>E I F" 
388 
by (auto simp: extensional_def PiE_def) 

389 
with \<open>Pi\<^sub>E I F = {}\<close> show False 

390 
by auto 

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391 
qed 
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392 
qed (auto simp: PiE_def) 
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393 

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394 
lemma PiE_arb: "f \<in> PiE S T \<Longrightarrow> x \<notin> S \<Longrightarrow> f x = undefined" 
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395 
unfolding PiE_def by auto (auto dest!: extensional_arb) 
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396 

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397 
lemma PiE_mem: "f \<in> PiE S T \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<in> T x" 
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398 
unfolding PiE_def by auto 
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399 

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400 
lemma PiE_fun_upd: "y \<in> T x \<Longrightarrow> f \<in> PiE S T \<Longrightarrow> f(x := y) \<in> PiE (insert x S) T" 
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401 
unfolding PiE_def extensional_def by auto 
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402 

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403 
lemma fun_upd_in_PiE: "x \<notin> S \<Longrightarrow> f \<in> PiE (insert x S) T \<Longrightarrow> f(x := undefined) \<in> PiE S T" 
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404 
unfolding PiE_def extensional_def by auto 
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405 

59425  406 
lemma PiE_insert_eq: "PiE (insert x S) T = (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)" 
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407 
proof  
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408 
{ 
59425  409 
fix f assume "f \<in> PiE (insert x S) T" "x \<notin> S" 
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410 
with assms have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)" 
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411 
by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem) 
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412 
} 
59425  413 
moreover 
414 
{ 

415 
fix f assume "f \<in> PiE (insert x S) T" "x \<in> S" 

416 
with assms have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)" 

417 
by (auto intro!: image_eqI[where x="(f x, f)"] intro: fun_upd_in_PiE PiE_mem simp: insert_absorb) 

418 
} 

419 
ultimately show ?thesis 

58783  420 
using assms by (auto intro: PiE_fun_upd) 
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421 
qed 
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422 

58783  423 
lemma PiE_Int: "Pi\<^sub>E I A \<inter> Pi\<^sub>E I B = Pi\<^sub>E I (\<lambda>x. A x \<inter> B x)" 
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424 
by (auto simp: PiE_def) 
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425 

58783  426 
lemma PiE_cong: "(\<And>i. i\<in>I \<Longrightarrow> A i = B i) \<Longrightarrow> Pi\<^sub>E I A = Pi\<^sub>E I B" 
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427 
unfolding PiE_def by (auto simp: Pi_cong) 
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428 

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429 
lemma PiE_E [elim]: 
58783  430 
assumes "f \<in> PiE A B" 
431 
obtains "x \<in> A" and "f x \<in> B x" 

432 
 "x \<notin> A" and "f x = undefined" 

433 
using assms by (auto simp: Pi_def PiE_def extensional_def) 

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434 

58783  435 
lemma PiE_I[intro!]: 
436 
"(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> (\<And>x. x \<notin> A \<Longrightarrow> f x = undefined) \<Longrightarrow> f \<in> PiE A B" 

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437 
by (simp add: PiE_def extensional_def) 
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438 

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439 
lemma PiE_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> PiE A B \<subseteq> PiE A C" 
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440 
by auto 
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441 

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442 
lemma PiE_iff: "f \<in> PiE I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I" 
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443 
by (simp add: PiE_def Pi_iff) 
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444 

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445 
lemma PiE_restrict[simp]: "f \<in> PiE A B \<Longrightarrow> restrict f A = f" 
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446 
by (simp add: extensional_restrict PiE_def) 
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447 

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448 
lemma restrict_PiE[simp]: "restrict f I \<in> PiE I S \<longleftrightarrow> f \<in> Pi I S" 
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449 
by (auto simp: PiE_iff) 
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450 

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451 
lemma PiE_eq_subset: 
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452 
assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}" 
58783  453 
and eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'" 
454 
and "i \<in> I" 

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455 
shows "F i \<subseteq> F' i" 
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456 
proof 
58783  457 
fix x 
458 
assume "x \<in> F i" 

459 
with ne have "\<forall>j. \<exists>y. (j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined)" 

53381  460 
by auto 
461 
from choice[OF this] obtain f 

462 
where f: " \<forall>j. (j \<in> I \<longrightarrow> f j \<in> F j \<and> (i = j \<longrightarrow> x = f j)) \<and> (j \<notin> I \<longrightarrow> f j = undefined)" .. 

58783  463 
then have "f \<in> Pi\<^sub>E I F" 
464 
by (auto simp: extensional_def PiE_def) 

465 
then have "f \<in> Pi\<^sub>E I F'" 

466 
using assms by simp 

467 
then show "x \<in> F' i" 

468 
using f \<open>i \<in> I\<close> by (auto simp: PiE_def) 

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469 
qed 
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470 

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471 
lemma PiE_eq_iff_not_empty: 
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472 
assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}" 
53015
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473 
shows "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)" 
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474 
proof (intro iffI ballI) 
58783  475 
fix i 
476 
assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'" 

477 
assume i: "i \<in> I" 

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478 
show "F i = F' i" 
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479 
using PiE_eq_subset[of I F F', OF ne eq i] 
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480 
using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i] 
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481 
by auto 
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482 
qed (auto simp: PiE_def) 
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483 

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484 
lemma PiE_eq_iff: 
53015
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485 
"Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))" 
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486 
proof (intro iffI disjCI) 
53015
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487 
assume eq[simp]: "Pi\<^sub>E I F = Pi\<^sub>E I F'" 
50123
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488 
assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))" 
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489 
then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})" 
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490 
using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto 
58783  491 
with PiE_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" 
492 
by auto 

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493 
next 
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494 
assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})" 
53015
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495 
then show "Pi\<^sub>E I F = Pi\<^sub>E I F'" 
50123
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496 
using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def) 
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497 
qed 
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498 

58783  499 
lemma extensional_funcset_fun_upd_restricts_rangeI: 
500 
"\<forall>y \<in> S. f x \<noteq> f y \<Longrightarrow> f \<in> (insert x S) \<rightarrow>\<^sub>E T \<Longrightarrow> f(x := undefined) \<in> S \<rightarrow>\<^sub>E (T  {f x})" 

50123
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501 
unfolding extensional_funcset_def extensional_def 
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changeset

502 
apply auto 
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503 
apply (case_tac "x = xa") 
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changeset

504 
apply auto 
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changeset

505 
done 
40631
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bulwahn
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changeset

506 

b3f85ba3dae4
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507 
lemma extensional_funcset_fun_upd_extends_rangeI: 
53015
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508 
assumes "a \<in> T" "f \<in> S \<rightarrow>\<^sub>E (T  {a})" 
58783  509 
shows "f(x := a) \<in> insert x S \<rightarrow>\<^sub>E T" 
40631
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changeset

510 
using assms unfolding extensional_funcset_def extensional_def by auto 
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511 

58783  512 

513 
subsubsection \<open>Injective Extensional Function Spaces\<close> 

40631
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514 

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515 
lemma extensional_funcset_fun_upd_inj_onI: 
58783  516 
assumes "f \<in> S \<rightarrow>\<^sub>E (T  {a})" 
517 
and "inj_on f S" 

40631
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518 
shows "inj_on (f(x := a)) S" 
58783  519 
using assms 
520 
unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI) 

40631
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521 

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522 
lemma extensional_funcset_extend_domain_inj_on_eq: 
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523 
assumes "x \<notin> S" 
58783  524 
shows "{f. f \<in> (insert x S) \<rightarrow>\<^sub>E T \<and> inj_on f (insert x S)} = 
525 
(\<lambda>(y, g). g(x:=y)) ` {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T  {y}) \<and> inj_on g S}" 

526 
using assms 

527 
apply (auto del: PiE_I PiE_E) 

528 
apply (auto intro: extensional_funcset_fun_upd_inj_onI 

529 
extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E) 

530 
apply (auto simp add: image_iff inj_on_def) 

531 
apply (rule_tac x="xa x" in exI) 

532 
apply (auto intro: PiE_mem del: PiE_I PiE_E) 

533 
apply (rule_tac x="xa(x := undefined)" in exI) 

534 
apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI) 

535 
apply (auto dest!: PiE_mem split: split_if_asm) 

536 
done 

40631
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bulwahn
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changeset

537 

b3f85ba3dae4
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538 
lemma extensional_funcset_extend_domain_inj_onI: 
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539 
assumes "x \<notin> S" 
53015
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wenzelm
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50123
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changeset

540 
shows "inj_on (\<lambda>(y, g). g(x := y)) {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T  {y}) \<and> inj_on g S}" 
58783  541 
using assms 
542 
apply (auto intro!: inj_onI) 

543 
apply (metis fun_upd_same) 

544 
apply (metis assms PiE_arb fun_upd_triv fun_upd_upd) 

545 
done 

40631
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changeset

546 

b3f85ba3dae4
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bulwahn
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547 

58783  548 
subsubsection \<open>Cardinality\<close> 
549 

550 
lemma finite_PiE: "finite S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> finite (T i)) \<Longrightarrow> finite (\<Pi>\<^sub>E i \<in> S. T i)" 

50123
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551 
by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq) 
69b35a75caf3
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hoelzl
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552 

53015
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diff
changeset

553 
lemma inj_combinator: "x \<notin> S \<Longrightarrow> inj_on (\<lambda>(y, g). g(x := y)) (T x \<times> Pi\<^sub>E S T)" 
50123
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554 
proof (safe intro!: inj_onI ext) 
58783  555 
fix f y g z 
556 
assume "x \<notin> S" 

557 
assume fg: "f \<in> Pi\<^sub>E S T" "g \<in> Pi\<^sub>E S T" 

50123
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changeset

558 
assume "f(x := y) = g(x := z)" 
69b35a75caf3
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559 
then have *: "\<And>i. (f(x := y)) i = (g(x := z)) i" 
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changeset

560 
unfolding fun_eq_iff by auto 
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changeset

561 
from this[of x] show "y = z" by simp 
58783  562 
fix i from *[of i] \<open>x \<notin> S\<close> fg show "f i = g i" 
50123
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changeset

563 
by (auto split: split_if_asm simp: PiE_def extensional_def) 
40631
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diff
changeset

564 
qed 
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changeset

565 

58783  566 
lemma card_PiE: "finite S \<Longrightarrow> card (\<Pi>\<^sub>E i \<in> S. T i) = (\<Prod> i\<in>S. card (T i))" 
50123
69b35a75caf3
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hoelzl
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changeset

567 
proof (induct rule: finite_induct) 
58783  568 
case empty 
569 
then show ?case by auto 

50123
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hoelzl
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diff
changeset

570 
next 
58783  571 
case (insert x S) 
572 
then show ?case 

50123
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merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
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diff
changeset

573 
by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product) 
40631
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bulwahn
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diff
changeset

574 
qed 
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changeset

575 

13586  576 
end 