src/HOL/Library/FuncSet.thy
author wenzelm
Fri Oct 09 20:26:03 2015 +0200 (2015-10-09)
changeset 61378 3e04c9ca001a
parent 61359 e985b52c3eb3
child 61384 9f5145281888
permissions -rw-r--r--
discontinued specific HTML syntax;
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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 non-empty @{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 Pi-sets 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 Pi-sets 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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subsection \<open>Bijections Between Sets\<close>
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text \<open>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}.\<close>
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lemma bij_betwI:
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  assumes "f \<in> A \<rightarrow> B"
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    and "g \<in> B \<rightarrow> A"
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    and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x"
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    and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y"
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  shows "bij_betw f A B"
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  unfolding bij_betw_def
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proof
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  show "inj_on f A"
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    by (metis g_f inj_on_def)
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  have "f ` A \<subseteq> B"
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    using \<open>f \<in> A \<rightarrow> B\<close> by auto
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  moreover
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  have "B \<subseteq> f ` A"
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    by auto (metis Pi_mem \<open>g \<in> B \<rightarrow> A\<close> f_g image_iff)
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  ultimately show "f ` A = B"
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    by blast
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qed
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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)
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lemma inj_on_compose: "bij_betw f A B \<Longrightarrow> inj_on g B \<Longrightarrow> 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: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> 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]: "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 \<open>Extensionality\<close>
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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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lemma extensional_arb: "f \<in> extensional A \<Longrightarrow> x \<notin> A \<Longrightarrow> 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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   281
  by (simp add: compose_def)
paulson@14853
   282
paulson@14853
   283
lemma extensionalityI:
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   284
  assumes "f \<in> extensional A"
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   285
    and "g \<in> extensional A"
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   286
    and "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
wenzelm@58783
   287
  shows "f = g"
wenzelm@58783
   288
  using assms by (force simp add: fun_eq_iff extensional_def)
paulson@14853
   289
nipkow@39595
   290
lemma extensional_restrict:  "f \<in> extensional A \<Longrightarrow> restrict f A = f"
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   291
  by (rule extensionalityI[OF restrict_extensional]) auto
nipkow@39595
   292
hoelzl@50123
   293
lemma extensional_subset: "f \<in> extensional A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f \<in> extensional B"
hoelzl@50123
   294
  unfolding extensional_def by auto
hoelzl@50123
   295
wenzelm@58783
   296
lemma inv_into_funcset: "f ` A = B \<Longrightarrow> (\<lambda>x\<in>B. inv_into A f x) \<in> B \<rightarrow> A"
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   297
  by (unfold inv_into_def) (fast intro: someI2)
paulson@14853
   298
wenzelm@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)"
wenzelm@58783
   300
  apply (simp add: bij_betw_def compose_def)
wenzelm@58783
   301
  apply (rule restrict_ext, auto)
wenzelm@58783
   302
  done
paulson@14853
   303
wenzelm@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)"
wenzelm@58783
   305
  apply (simp add: compose_def)
wenzelm@58783
   306
  apply (rule restrict_ext)
wenzelm@58783
   307
  apply (simp add: f_inv_into_f)
wenzelm@58783
   308
  done
paulson@14853
   309
hoelzl@50123
   310
lemma extensional_insert[intro, simp]:
hoelzl@50123
   311
  assumes "a \<in> extensional (insert i I)"
hoelzl@50123
   312
  shows "a(i := b) \<in> extensional (insert i I)"
hoelzl@50123
   313
  using assms unfolding extensional_def by auto
hoelzl@50123
   314
wenzelm@58783
   315
lemma extensional_Int[simp]: "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
hoelzl@50123
   316
  unfolding extensional_def by auto
hoelzl@50123
   317
hoelzl@50123
   318
lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
hoelzl@50123
   319
  by (auto simp: extensional_def)
hoelzl@50123
   320
hoelzl@50123
   321
lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
hoelzl@50123
   322
  unfolding restrict_def extensional_def by auto
hoelzl@50123
   323
hoelzl@50123
   324
lemma extensional_insert_undefined[intro, simp]:
hoelzl@50123
   325
  "a \<in> extensional (insert i I) \<Longrightarrow> a(i := undefined) \<in> extensional I"
hoelzl@50123
   326
  unfolding extensional_def by auto
hoelzl@50123
   327
hoelzl@50123
   328
lemma extensional_insert_cancel[intro, simp]:
hoelzl@50123
   329
  "a \<in> extensional I \<Longrightarrow> a \<in> extensional (insert i I)"
hoelzl@50123
   330
  unfolding extensional_def by auto
hoelzl@50123
   331
paulson@14762
   332
wenzelm@58783
   333
subsection \<open>Cardinality\<close>
paulson@14745
   334
wenzelm@58783
   335
lemma card_inj: "f \<in> A \<rightarrow> B \<Longrightarrow> inj_on f A \<Longrightarrow> finite B \<Longrightarrow> card A \<le> card B"
wenzelm@58783
   336
  by (rule card_inj_on_le) auto
paulson@14745
   337
paulson@14745
   338
lemma card_bij:
wenzelm@58783
   339
  assumes "f \<in> A \<rightarrow> B" "inj_on f A"
wenzelm@58783
   340
    and "g \<in> B \<rightarrow> A" "inj_on g B"
wenzelm@58783
   341
    and "finite A" "finite B"
wenzelm@58783
   342
  shows "card A = card B"
wenzelm@58783
   343
  using assms by (blast intro: card_inj order_antisym)
paulson@14745
   344
wenzelm@58783
   345
wenzelm@58783
   346
subsection \<open>Extensional Function Spaces\<close>
bulwahn@40631
   347
wenzelm@58783
   348
definition PiE :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
wenzelm@58783
   349
  where "PiE S T = Pi S T \<inter> extensional S"
hoelzl@50123
   350
wenzelm@53015
   351
abbreviation "Pi\<^sub>E A B \<equiv> PiE A B"
bulwahn@40631
   352
wenzelm@58783
   353
syntax
wenzelm@58783
   354
  "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3PIE _:_./ _)" 10)
wenzelm@58783
   355
syntax (xsymbols)
wenzelm@58783
   356
  "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
wenzelm@58783
   357
translations "\<Pi>\<^sub>E x\<in>A. B" \<rightleftharpoons> "CONST Pi\<^sub>E A (\<lambda>x. B)"
hoelzl@50123
   358
wenzelm@58783
   359
abbreviation extensional_funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "->\<^sub>E" 60)
wenzelm@58783
   360
  where "A ->\<^sub>E B \<equiv> (\<Pi>\<^sub>E i\<in>A. B)"
hoelzl@50123
   361
hoelzl@50123
   362
notation (xsymbols)
wenzelm@53015
   363
  extensional_funcset  (infixr "\<rightarrow>\<^sub>E" 60)
bulwahn@40631
   364
wenzelm@58783
   365
lemma extensional_funcset_def: "extensional_funcset S T = (S \<rightarrow> T) \<inter> extensional S"
hoelzl@50123
   366
  by (simp add: PiE_def)
hoelzl@50123
   367
wenzelm@58783
   368
lemma PiE_empty_domain[simp]: "PiE {} T = {\<lambda>x. undefined}"
hoelzl@50123
   369
  unfolding PiE_def by simp
hoelzl@50123
   370
hoelzl@54417
   371
lemma PiE_UNIV_domain: "PiE UNIV T = Pi UNIV T"
hoelzl@54417
   372
  unfolding PiE_def by simp
hoelzl@54417
   373
wenzelm@58783
   374
lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (\<Pi>\<^sub>E i\<in>I. F i) = {}"
hoelzl@50123
   375
  unfolding PiE_def by auto
bulwahn@40631
   376
wenzelm@58783
   377
lemma PiE_eq_empty_iff: "Pi\<^sub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
hoelzl@50123
   378
proof
wenzelm@53015
   379
  assume "Pi\<^sub>E I F = {}"
hoelzl@50123
   380
  show "\<exists>i\<in>I. F i = {}"
hoelzl@50123
   381
  proof (rule ccontr)
hoelzl@50123
   382
    assume "\<not> ?thesis"
wenzelm@58783
   383
    then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)"
wenzelm@58783
   384
      by auto
wenzelm@53381
   385
    from choice[OF this]
wenzelm@53381
   386
    obtain f where " \<forall>x. (x \<in> I \<longrightarrow> f x \<in> F x) \<and> (x \<notin> I \<longrightarrow> f x = undefined)" ..
wenzelm@58783
   387
    then have "f \<in> Pi\<^sub>E I F"
wenzelm@58783
   388
      by (auto simp: extensional_def PiE_def)
wenzelm@58783
   389
    with \<open>Pi\<^sub>E I F = {}\<close> show False
wenzelm@58783
   390
      by auto
hoelzl@50123
   391
  qed
hoelzl@50123
   392
qed (auto simp: PiE_def)
bulwahn@40631
   393
hoelzl@50123
   394
lemma PiE_arb: "f \<in> PiE S T \<Longrightarrow> x \<notin> S \<Longrightarrow> f x = undefined"
hoelzl@50123
   395
  unfolding PiE_def by auto (auto dest!: extensional_arb)
hoelzl@50123
   396
hoelzl@50123
   397
lemma PiE_mem: "f \<in> PiE S T \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<in> T x"
hoelzl@50123
   398
  unfolding PiE_def by auto
bulwahn@40631
   399
hoelzl@50123
   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"
hoelzl@50123
   401
  unfolding PiE_def extensional_def by auto
bulwahn@40631
   402
hoelzl@50123
   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"
hoelzl@50123
   404
  unfolding PiE_def extensional_def by auto
hoelzl@50123
   405
hoelzl@59425
   406
lemma PiE_insert_eq: "PiE (insert x S) T = (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)"
bulwahn@40631
   407
proof -
bulwahn@40631
   408
  {
hoelzl@59425
   409
    fix f assume "f \<in> PiE (insert x S) T" "x \<notin> S"
hoelzl@50123
   410
    with assms have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)"
hoelzl@50123
   411
      by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem)
bulwahn@40631
   412
  }
hoelzl@59425
   413
  moreover
hoelzl@59425
   414
  {
hoelzl@59425
   415
    fix f assume "f \<in> PiE (insert x S) T" "x \<in> S"
hoelzl@59425
   416
    with assms have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)"
hoelzl@59425
   417
      by (auto intro!: image_eqI[where x="(f x, f)"] intro: fun_upd_in_PiE PiE_mem simp: insert_absorb)
hoelzl@59425
   418
  }
hoelzl@59425
   419
  ultimately show ?thesis
wenzelm@58783
   420
    using assms by (auto intro: PiE_fun_upd)
bulwahn@40631
   421
qed
bulwahn@40631
   422
wenzelm@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)"
hoelzl@50123
   424
  by (auto simp: PiE_def)
hoelzl@50123
   425
wenzelm@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"
hoelzl@50123
   427
  unfolding PiE_def by (auto simp: Pi_cong)
hoelzl@50123
   428
hoelzl@50123
   429
lemma PiE_E [elim]:
wenzelm@58783
   430
  assumes "f \<in> PiE A B"
wenzelm@58783
   431
  obtains "x \<in> A" and "f x \<in> B x"
wenzelm@58783
   432
    | "x \<notin> A" and "f x = undefined"
wenzelm@58783
   433
  using assms by (auto simp: Pi_def PiE_def extensional_def)
hoelzl@50123
   434
wenzelm@58783
   435
lemma PiE_I[intro!]:
wenzelm@58783
   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"
hoelzl@50123
   437
  by (simp add: PiE_def extensional_def)
hoelzl@50123
   438
hoelzl@50123
   439
lemma PiE_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> PiE A B \<subseteq> PiE A C"
hoelzl@50123
   440
  by auto
hoelzl@50123
   441
hoelzl@50123
   442
lemma PiE_iff: "f \<in> PiE I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I"
hoelzl@50123
   443
  by (simp add: PiE_def Pi_iff)
hoelzl@50123
   444
hoelzl@50123
   445
lemma PiE_restrict[simp]:  "f \<in> PiE A B \<Longrightarrow> restrict f A = f"
hoelzl@50123
   446
  by (simp add: extensional_restrict PiE_def)
hoelzl@50123
   447
hoelzl@50123
   448
lemma restrict_PiE[simp]: "restrict f I \<in> PiE I S \<longleftrightarrow> f \<in> Pi I S"
hoelzl@50123
   449
  by (auto simp: PiE_iff)
hoelzl@50123
   450
hoelzl@50123
   451
lemma PiE_eq_subset:
hoelzl@50123
   452
  assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
wenzelm@58783
   453
    and eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
wenzelm@58783
   454
    and "i \<in> I"
hoelzl@50123
   455
  shows "F i \<subseteq> F' i"
hoelzl@50123
   456
proof
wenzelm@58783
   457
  fix x
wenzelm@58783
   458
  assume "x \<in> F i"
wenzelm@58783
   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)"
wenzelm@53381
   460
    by auto
wenzelm@53381
   461
  from choice[OF this] obtain f
wenzelm@53381
   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)" ..
wenzelm@58783
   463
  then have "f \<in> Pi\<^sub>E I F"
wenzelm@58783
   464
    by (auto simp: extensional_def PiE_def)
wenzelm@58783
   465
  then have "f \<in> Pi\<^sub>E I F'"
wenzelm@58783
   466
    using assms by simp
wenzelm@58783
   467
  then show "x \<in> F' i"
wenzelm@58783
   468
    using f \<open>i \<in> I\<close> by (auto simp: PiE_def)
hoelzl@50123
   469
qed
hoelzl@50123
   470
hoelzl@50123
   471
lemma PiE_eq_iff_not_empty:
hoelzl@50123
   472
  assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
wenzelm@53015
   473
  shows "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
hoelzl@50123
   474
proof (intro iffI ballI)
wenzelm@58783
   475
  fix i
wenzelm@58783
   476
  assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
wenzelm@58783
   477
  assume i: "i \<in> I"
hoelzl@50123
   478
  show "F i = F' i"
hoelzl@50123
   479
    using PiE_eq_subset[of I F F', OF ne eq i]
hoelzl@50123
   480
    using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
hoelzl@50123
   481
    by auto
hoelzl@50123
   482
qed (auto simp: PiE_def)
hoelzl@50123
   483
hoelzl@50123
   484
lemma PiE_eq_iff:
wenzelm@53015
   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 = {}))"
hoelzl@50123
   486
proof (intro iffI disjCI)
wenzelm@53015
   487
  assume eq[simp]: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
hoelzl@50123
   488
  assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
hoelzl@50123
   489
  then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
hoelzl@50123
   490
    using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto
wenzelm@58783
   491
  with PiE_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i"
wenzelm@58783
   492
    by auto
hoelzl@50123
   493
next
hoelzl@50123
   494
  assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
wenzelm@53015
   495
  then show "Pi\<^sub>E I F = Pi\<^sub>E I F'"
hoelzl@50123
   496
    using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def)
hoelzl@50123
   497
qed
hoelzl@50123
   498
wenzelm@58783
   499
lemma extensional_funcset_fun_upd_restricts_rangeI:
wenzelm@58783
   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})"
hoelzl@50123
   501
  unfolding extensional_funcset_def extensional_def
hoelzl@50123
   502
  apply auto
hoelzl@50123
   503
  apply (case_tac "x = xa")
hoelzl@50123
   504
  apply auto
hoelzl@50123
   505
  done
bulwahn@40631
   506
bulwahn@40631
   507
lemma extensional_funcset_fun_upd_extends_rangeI:
wenzelm@53015
   508
  assumes "a \<in> T" "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
wenzelm@58783
   509
  shows "f(x := a) \<in> insert x S \<rightarrow>\<^sub>E  T"
bulwahn@40631
   510
  using assms unfolding extensional_funcset_def extensional_def by auto
bulwahn@40631
   511
wenzelm@58783
   512
wenzelm@58783
   513
subsubsection \<open>Injective Extensional Function Spaces\<close>
bulwahn@40631
   514
bulwahn@40631
   515
lemma extensional_funcset_fun_upd_inj_onI:
wenzelm@58783
   516
  assumes "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
wenzelm@58783
   517
    and "inj_on f S"
bulwahn@40631
   518
  shows "inj_on (f(x := a)) S"
wenzelm@58783
   519
  using assms
wenzelm@58783
   520
  unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
bulwahn@40631
   521
bulwahn@40631
   522
lemma extensional_funcset_extend_domain_inj_on_eq:
bulwahn@40631
   523
  assumes "x \<notin> S"
wenzelm@58783
   524
  shows "{f. f \<in> (insert x S) \<rightarrow>\<^sub>E T \<and> inj_on f (insert x S)} =
wenzelm@58783
   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}"
wenzelm@58783
   526
  using assms
wenzelm@58783
   527
  apply (auto del: PiE_I PiE_E)
wenzelm@58783
   528
  apply (auto intro: extensional_funcset_fun_upd_inj_onI
wenzelm@58783
   529
    extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
wenzelm@58783
   530
  apply (auto simp add: image_iff inj_on_def)
wenzelm@58783
   531
  apply (rule_tac x="xa x" in exI)
wenzelm@58783
   532
  apply (auto intro: PiE_mem del: PiE_I PiE_E)
wenzelm@58783
   533
  apply (rule_tac x="xa(x := undefined)" in exI)
wenzelm@58783
   534
  apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
wenzelm@58783
   535
  apply (auto dest!: PiE_mem split: split_if_asm)
wenzelm@58783
   536
  done
bulwahn@40631
   537
bulwahn@40631
   538
lemma extensional_funcset_extend_domain_inj_onI:
bulwahn@40631
   539
  assumes "x \<notin> S"
wenzelm@53015
   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}"
wenzelm@58783
   541
  using assms
wenzelm@58783
   542
  apply (auto intro!: inj_onI)
wenzelm@58783
   543
  apply (metis fun_upd_same)
wenzelm@58783
   544
  apply (metis assms PiE_arb fun_upd_triv fun_upd_upd)
wenzelm@58783
   545
  done
bulwahn@40631
   546
bulwahn@40631
   547
wenzelm@58783
   548
subsubsection \<open>Cardinality\<close>
wenzelm@58783
   549
wenzelm@58783
   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)"
hoelzl@50123
   551
  by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq)
hoelzl@50123
   552
wenzelm@53015
   553
lemma inj_combinator: "x \<notin> S \<Longrightarrow> inj_on (\<lambda>(y, g). g(x := y)) (T x \<times> Pi\<^sub>E S T)"
hoelzl@50123
   554
proof (safe intro!: inj_onI ext)
wenzelm@58783
   555
  fix f y g z
wenzelm@58783
   556
  assume "x \<notin> S"
wenzelm@58783
   557
  assume fg: "f \<in> Pi\<^sub>E S T" "g \<in> Pi\<^sub>E S T"
hoelzl@50123
   558
  assume "f(x := y) = g(x := z)"
hoelzl@50123
   559
  then have *: "\<And>i. (f(x := y)) i = (g(x := z)) i"
hoelzl@50123
   560
    unfolding fun_eq_iff by auto
hoelzl@50123
   561
  from this[of x] show "y = z" by simp
wenzelm@58783
   562
  fix i from *[of i] \<open>x \<notin> S\<close> fg show "f i = g i"
hoelzl@50123
   563
    by (auto split: split_if_asm simp: PiE_def extensional_def)
bulwahn@40631
   564
qed
bulwahn@40631
   565
wenzelm@58783
   566
lemma card_PiE: "finite S \<Longrightarrow> card (\<Pi>\<^sub>E i \<in> S. T i) = (\<Prod> i\<in>S. card (T i))"
hoelzl@50123
   567
proof (induct rule: finite_induct)
wenzelm@58783
   568
  case empty
wenzelm@58783
   569
  then show ?case by auto
hoelzl@50123
   570
next
wenzelm@58783
   571
  case (insert x S)
wenzelm@58783
   572
  then show ?case
hoelzl@50123
   573
    by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)
bulwahn@40631
   574
qed
bulwahn@40631
   575
paulson@13586
   576
end