src/HOL/Library/FuncSet.thy
author hoelzl
Tue Nov 12 19:28:56 2013 +0100 (2013-11-12)
changeset 54417 dbb8ecfe1337
parent 53381 355a4cac5440
child 56777 9c3f0ae99532
permissions -rw-r--r--
add restrict_space measure
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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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header {* Pi and Function Sets *}
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theory FuncSet
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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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  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[intro!]: "(!!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 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]:
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  "f : Pi A B ==> (f x : B x ==> Q) ==> (x ~: A ==> Q) ==> Q"
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by(auto simp: Pi_def)
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lemma Pi_cong:
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  "(\<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 -> 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
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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: 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 non-empty @{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_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" 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 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" 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)" 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 `finite I` 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 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))" by auto
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qed auto
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lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
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by (simp add: Pi_def)
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text{*Covariance of Pi-sets 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 auto
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text{*Contravariance of Pi-sets 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 auto
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lemma prod_final:
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  assumes 1: "fst \<circ> f \<in> Pi A B" 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 "...  \<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{*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: fun_eq_iff 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: "(\<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:
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    "(\<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 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: fun_eq_iff 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: 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]:
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  "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{*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_betwI:
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assumes "f \<in> A \<rightarrow> B" and "g \<in> B \<rightarrow> A"
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    and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x" 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" by (metis g_f inj_on_def)
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next
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  have "f ` A \<subseteq> B" using `f \<in> A \<rightarrow> B` by auto
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  moreover
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  have "B \<subseteq> f ` A" by auto (metis Pi_mem `g \<in> B \<rightarrow> A` f_g image_iff)
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  ultimately show "f ` A = B" 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:
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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_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; 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: fun_eq_iff extensional_def)
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lemma extensional_restrict:  "f \<in> extensional A \<Longrightarrow> restrict f A = f"
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by(rule extensionalityI[OF restrict_extensional]) auto
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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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lemma inv_into_funcset: "f ` A = B ==> (\<lambda>x\<in>B. inv_into A f x) : B -> A"
nipkow@33057
   293
by (unfold inv_into_def) (fast intro: someI2)
paulson@14853
   294
nipkow@33057
   295
lemma compose_inv_into_id:
nipkow@33057
   296
  "bij_betw f A B ==> compose A (\<lambda>y\<in>B. inv_into A f y) f = (\<lambda>x\<in>A. x)"
nipkow@31754
   297
apply (simp add: bij_betw_def compose_def)
nipkow@31754
   298
apply (rule restrict_ext, auto)
nipkow@31754
   299
done
paulson@14853
   300
nipkow@33057
   301
lemma compose_id_inv_into:
nipkow@33057
   302
  "f ` A = B ==> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
nipkow@31754
   303
apply (simp add: compose_def)
nipkow@31754
   304
apply (rule restrict_ext)
nipkow@33057
   305
apply (simp add: f_inv_into_f)
nipkow@31754
   306
done
paulson@14853
   307
hoelzl@50123
   308
lemma extensional_insert[intro, simp]:
hoelzl@50123
   309
  assumes "a \<in> extensional (insert i I)"
hoelzl@50123
   310
  shows "a(i := b) \<in> extensional (insert i I)"
hoelzl@50123
   311
  using assms unfolding extensional_def by auto
hoelzl@50123
   312
hoelzl@50123
   313
lemma extensional_Int[simp]:
hoelzl@50123
   314
  "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
hoelzl@50123
   315
  unfolding extensional_def by auto
hoelzl@50123
   316
hoelzl@50123
   317
lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
hoelzl@50123
   318
  by (auto simp: extensional_def)
hoelzl@50123
   319
hoelzl@50123
   320
lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
hoelzl@50123
   321
  unfolding restrict_def extensional_def by auto
hoelzl@50123
   322
hoelzl@50123
   323
lemma extensional_insert_undefined[intro, simp]:
hoelzl@50123
   324
  "a \<in> extensional (insert i I) \<Longrightarrow> a(i := undefined) \<in> extensional I"
hoelzl@50123
   325
  unfolding extensional_def by auto
hoelzl@50123
   326
hoelzl@50123
   327
lemma extensional_insert_cancel[intro, simp]:
hoelzl@50123
   328
  "a \<in> extensional I \<Longrightarrow> a \<in> extensional (insert i I)"
hoelzl@50123
   329
  unfolding extensional_def by auto
hoelzl@50123
   330
paulson@14762
   331
paulson@14745
   332
subsection{*Cardinality*}
paulson@14745
   333
paulson@14745
   334
lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
nipkow@31754
   335
by (rule card_inj_on_le) auto
paulson@14745
   336
paulson@14745
   337
lemma card_bij:
nipkow@31754
   338
  "[|f \<in> A\<rightarrow>B; inj_on f A;
nipkow@31754
   339
     g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
nipkow@31754
   340
by (blast intro: card_inj order_antisym)
paulson@14745
   341
bulwahn@40631
   342
subsection {* Extensional Function Spaces *} 
bulwahn@40631
   343
hoelzl@50123
   344
definition PiE :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set" where
hoelzl@50123
   345
  "PiE S T = Pi S T \<inter> extensional S"
hoelzl@50123
   346
wenzelm@53015
   347
abbreviation "Pi\<^sub>E A B \<equiv> PiE A B"
bulwahn@40631
   348
hoelzl@50123
   349
syntax "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
hoelzl@50123
   350
wenzelm@53015
   351
syntax (xsymbols) "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
hoelzl@50123
   352
wenzelm@53015
   353
syntax (HTML output) "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
hoelzl@50123
   354
wenzelm@53015
   355
translations "PIE x:A. B" == "CONST Pi\<^sub>E A (%x. B)"
bulwahn@40631
   356
wenzelm@53015
   357
abbreviation extensional_funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "->\<^sub>E" 60) where
wenzelm@53015
   358
  "A ->\<^sub>E B \<equiv> (\<Pi>\<^sub>E i\<in>A. B)"
hoelzl@50123
   359
hoelzl@50123
   360
notation (xsymbols)
wenzelm@53015
   361
  extensional_funcset  (infixr "\<rightarrow>\<^sub>E" 60)
bulwahn@40631
   362
hoelzl@50123
   363
lemma extensional_funcset_def: "extensional_funcset S T = (S -> T) \<inter> extensional S"
hoelzl@50123
   364
  by (simp add: PiE_def)
hoelzl@50123
   365
hoelzl@50123
   366
lemma PiE_empty_domain[simp]: "PiE {} T = {%x. undefined}"
hoelzl@50123
   367
  unfolding PiE_def by simp
hoelzl@50123
   368
hoelzl@54417
   369
lemma PiE_UNIV_domain: "PiE UNIV T = Pi UNIV T"
hoelzl@54417
   370
  unfolding PiE_def by simp
hoelzl@54417
   371
hoelzl@50123
   372
lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (PIE i:I. F i) = {}"
hoelzl@50123
   373
  unfolding PiE_def by auto
bulwahn@40631
   374
hoelzl@50123
   375
lemma PiE_eq_empty_iff:
wenzelm@53015
   376
  "Pi\<^sub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
hoelzl@50123
   377
proof
wenzelm@53015
   378
  assume "Pi\<^sub>E I F = {}"
hoelzl@50123
   379
  show "\<exists>i\<in>I. F i = {}"
hoelzl@50123
   380
  proof (rule ccontr)
hoelzl@50123
   381
    assume "\<not> ?thesis"
hoelzl@50123
   382
    then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto
wenzelm@53381
   383
    from choice[OF this]
wenzelm@53381
   384
    obtain f where " \<forall>x. (x \<in> I \<longrightarrow> f x \<in> F x) \<and> (x \<notin> I \<longrightarrow> f x = undefined)" ..
wenzelm@53015
   385
    then have "f \<in> Pi\<^sub>E I F" by (auto simp: extensional_def PiE_def)
wenzelm@53015
   386
    with `Pi\<^sub>E I F = {}` show False by auto
hoelzl@50123
   387
  qed
hoelzl@50123
   388
qed (auto simp: PiE_def)
bulwahn@40631
   389
hoelzl@50123
   390
lemma PiE_arb: "f \<in> PiE S T \<Longrightarrow> x \<notin> S \<Longrightarrow> f x = undefined"
hoelzl@50123
   391
  unfolding PiE_def by auto (auto dest!: extensional_arb)
hoelzl@50123
   392
hoelzl@50123
   393
lemma PiE_mem: "f \<in> PiE S T \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<in> T x"
hoelzl@50123
   394
  unfolding PiE_def by auto
bulwahn@40631
   395
hoelzl@50123
   396
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
   397
  unfolding PiE_def extensional_def by auto
bulwahn@40631
   398
hoelzl@50123
   399
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
   400
  unfolding PiE_def extensional_def by auto
hoelzl@50123
   401
hoelzl@50123
   402
lemma PiE_insert_eq:
bulwahn@40631
   403
  assumes "x \<notin> S"
hoelzl@50123
   404
  shows "PiE (insert x S) T = (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)"
bulwahn@40631
   405
proof -
bulwahn@40631
   406
  {
hoelzl@50123
   407
    fix f assume "f \<in> PiE (insert x S) T"
hoelzl@50123
   408
    with assms have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)"
hoelzl@50123
   409
      by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem)
bulwahn@40631
   410
  }
hoelzl@50123
   411
  then show ?thesis using assms by (auto intro: PiE_fun_upd)
bulwahn@40631
   412
qed
bulwahn@40631
   413
wenzelm@53015
   414
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
   415
  by (auto simp: PiE_def)
hoelzl@50123
   416
hoelzl@50123
   417
lemma PiE_cong:
wenzelm@53015
   418
  "(\<And>i. i\<in>I \<Longrightarrow> A i = B i) \<Longrightarrow> Pi\<^sub>E I A = Pi\<^sub>E I B"
hoelzl@50123
   419
  unfolding PiE_def by (auto simp: Pi_cong)
hoelzl@50123
   420
hoelzl@50123
   421
lemma PiE_E [elim]:
hoelzl@50123
   422
  "f \<in> PiE A B \<Longrightarrow> (x \<in> A \<Longrightarrow> f x \<in> B x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> f x = undefined \<Longrightarrow> Q) \<Longrightarrow> Q"
hoelzl@50123
   423
by(auto simp: Pi_def PiE_def extensional_def)
hoelzl@50123
   424
hoelzl@50123
   425
lemma PiE_I[intro!]: "(\<And>x. x \<in> A ==> f x \<in> B x) \<Longrightarrow> (\<And>x. x \<notin> A \<Longrightarrow> f x = undefined) \<Longrightarrow> f \<in> PiE A B"
hoelzl@50123
   426
  by (simp add: PiE_def extensional_def)
hoelzl@50123
   427
hoelzl@50123
   428
lemma PiE_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> PiE A B \<subseteq> PiE A C"
hoelzl@50123
   429
  by auto
hoelzl@50123
   430
hoelzl@50123
   431
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
   432
  by (simp add: PiE_def Pi_iff)
hoelzl@50123
   433
hoelzl@50123
   434
lemma PiE_restrict[simp]:  "f \<in> PiE A B \<Longrightarrow> restrict f A = f"
hoelzl@50123
   435
  by (simp add: extensional_restrict PiE_def)
hoelzl@50123
   436
hoelzl@50123
   437
lemma restrict_PiE[simp]: "restrict f I \<in> PiE I S \<longleftrightarrow> f \<in> Pi I S"
hoelzl@50123
   438
  by (auto simp: PiE_iff)
hoelzl@50123
   439
hoelzl@50123
   440
lemma PiE_eq_subset:
hoelzl@50123
   441
  assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
wenzelm@53015
   442
  assumes eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'" and "i \<in> I"
hoelzl@50123
   443
  shows "F i \<subseteq> F' i"
hoelzl@50123
   444
proof
hoelzl@50123
   445
  fix x assume "x \<in> F i"
wenzelm@53381
   446
  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
   447
    by auto
wenzelm@53381
   448
  from choice[OF this] obtain f
wenzelm@53381
   449
    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@53015
   450
  then have "f \<in> Pi\<^sub>E I F" by (auto simp: extensional_def PiE_def)
wenzelm@53015
   451
  then have "f \<in> Pi\<^sub>E I F'" using assms by simp
hoelzl@50123
   452
  then show "x \<in> F' i" using f `i \<in> I` by (auto simp: PiE_def)
hoelzl@50123
   453
qed
hoelzl@50123
   454
hoelzl@50123
   455
lemma PiE_eq_iff_not_empty:
hoelzl@50123
   456
  assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
wenzelm@53015
   457
  shows "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
hoelzl@50123
   458
proof (intro iffI ballI)
wenzelm@53015
   459
  fix i assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'" and i: "i \<in> I"
hoelzl@50123
   460
  show "F i = F' i"
hoelzl@50123
   461
    using PiE_eq_subset[of I F F', OF ne eq i]
hoelzl@50123
   462
    using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
hoelzl@50123
   463
    by auto
hoelzl@50123
   464
qed (auto simp: PiE_def)
hoelzl@50123
   465
hoelzl@50123
   466
lemma PiE_eq_iff:
wenzelm@53015
   467
  "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
   468
proof (intro iffI disjCI)
wenzelm@53015
   469
  assume eq[simp]: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
hoelzl@50123
   470
  assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
hoelzl@50123
   471
  then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
hoelzl@50123
   472
    using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto
hoelzl@50123
   473
  with PiE_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto
hoelzl@50123
   474
next
hoelzl@50123
   475
  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
   476
  then show "Pi\<^sub>E I F = Pi\<^sub>E I F'"
hoelzl@50123
   477
    using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def)
hoelzl@50123
   478
qed
hoelzl@50123
   479
hoelzl@50123
   480
lemma extensional_funcset_fun_upd_restricts_rangeI: 
wenzelm@53015
   481
  "\<forall>y \<in> S. f x \<noteq> f y \<Longrightarrow> f : (insert x S) \<rightarrow>\<^sub>E T ==> f(x := undefined) : S \<rightarrow>\<^sub>E (T - {f x})"
hoelzl@50123
   482
  unfolding extensional_funcset_def extensional_def
hoelzl@50123
   483
  apply auto
hoelzl@50123
   484
  apply (case_tac "x = xa")
hoelzl@50123
   485
  apply auto
hoelzl@50123
   486
  done
bulwahn@40631
   487
bulwahn@40631
   488
lemma extensional_funcset_fun_upd_extends_rangeI:
wenzelm@53015
   489
  assumes "a \<in> T" "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
wenzelm@53015
   490
  shows "f(x := a) \<in> (insert x S) \<rightarrow>\<^sub>E  T"
bulwahn@40631
   491
  using assms unfolding extensional_funcset_def extensional_def by auto
bulwahn@40631
   492
bulwahn@40631
   493
subsubsection {* Injective Extensional Function Spaces *}
bulwahn@40631
   494
bulwahn@40631
   495
lemma extensional_funcset_fun_upd_inj_onI:
wenzelm@53015
   496
  assumes "f \<in> S \<rightarrow>\<^sub>E (T - {a})" "inj_on f S"
bulwahn@40631
   497
  shows "inj_on (f(x := a)) S"
bulwahn@40631
   498
  using assms unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
bulwahn@40631
   499
bulwahn@40631
   500
lemma extensional_funcset_extend_domain_inj_on_eq:
bulwahn@40631
   501
  assumes "x \<notin> S"
wenzelm@53015
   502
  shows"{f. f \<in> (insert x S) \<rightarrow>\<^sub>E T \<and> inj_on f (insert x S)} =
wenzelm@53015
   503
    (%(y, g). g(x:=y)) ` {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T - {y}) \<and> inj_on g S}"
bulwahn@40631
   504
proof -
bulwahn@40631
   505
  from assms show ?thesis
hoelzl@50123
   506
    apply (auto del: PiE_I PiE_E)
hoelzl@50123
   507
    apply (auto intro: extensional_funcset_fun_upd_inj_onI extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
bulwahn@40631
   508
    apply (auto simp add: image_iff inj_on_def)
bulwahn@40631
   509
    apply (rule_tac x="xa x" in exI)
hoelzl@50123
   510
    apply (auto intro: PiE_mem del: PiE_I PiE_E)
bulwahn@40631
   511
    apply (rule_tac x="xa(x := undefined)" in exI)
bulwahn@40631
   512
    apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
hoelzl@50123
   513
    apply (auto dest!: PiE_mem split: split_if_asm)
bulwahn@40631
   514
    done
bulwahn@40631
   515
qed
bulwahn@40631
   516
bulwahn@40631
   517
lemma extensional_funcset_extend_domain_inj_onI:
bulwahn@40631
   518
  assumes "x \<notin> S"
wenzelm@53015
   519
  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}"
bulwahn@40631
   520
proof -
bulwahn@40631
   521
  from assms show ?thesis
bulwahn@40631
   522
    apply (auto intro!: inj_onI)
bulwahn@40631
   523
    apply (metis fun_upd_same)
hoelzl@50123
   524
    by (metis assms PiE_arb fun_upd_triv fun_upd_upd)
bulwahn@40631
   525
qed
bulwahn@40631
   526
  
bulwahn@40631
   527
bulwahn@40631
   528
subsubsection {* Cardinality *}
bulwahn@40631
   529
hoelzl@50123
   530
lemma finite_PiE: "finite S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> finite (T i)) \<Longrightarrow> finite (PIE i : S. T i)"
hoelzl@50123
   531
  by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq)
hoelzl@50123
   532
wenzelm@53015
   533
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
   534
proof (safe intro!: inj_onI ext)
wenzelm@53015
   535
  fix f y g z assume "x \<notin> S" and fg: "f \<in> Pi\<^sub>E S T" "g \<in> Pi\<^sub>E S T"
hoelzl@50123
   536
  assume "f(x := y) = g(x := z)"
hoelzl@50123
   537
  then have *: "\<And>i. (f(x := y)) i = (g(x := z)) i"
hoelzl@50123
   538
    unfolding fun_eq_iff by auto
hoelzl@50123
   539
  from this[of x] show "y = z" by simp
hoelzl@50123
   540
  fix i from *[of i] `x \<notin> S` fg show "f i = g i"
hoelzl@50123
   541
    by (auto split: split_if_asm simp: PiE_def extensional_def)
bulwahn@40631
   542
qed
bulwahn@40631
   543
hoelzl@50123
   544
lemma card_PiE:
hoelzl@50123
   545
  "finite S \<Longrightarrow> card (PIE i : S. T i) = (\<Prod> i\<in>S. card (T i))"
hoelzl@50123
   546
proof (induct rule: finite_induct)
hoelzl@50123
   547
  case empty then show ?case by auto
hoelzl@50123
   548
next
hoelzl@50123
   549
  case (insert x S) then show ?case
hoelzl@50123
   550
    by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)
bulwahn@40631
   551
qed
bulwahn@40631
   552
paulson@13586
   553
end