src/HOL/Library/FuncSet.thy
author hoelzl
Fri Nov 16 18:45:57 2012 +0100 (2012-11-16)
changeset 50104 de19856feb54
parent 47761 dfe747e72fa8
child 50123 69b35a75caf3
permissions -rw-r--r--
move theorems to be more generally useable
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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_UNIV [simp]: "A -> UNIV = UNIV"
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by (simp add: Pi_def)
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(*
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lemma funcset_id [simp]: "(%x. x): A -> A"
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  by (simp add: Pi_def)
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*)
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text{*Covariance of 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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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: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
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  by (simp add: Pi_def restrict_def)
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lemma restrictI[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
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  by (simp add: Pi_def restrict_def)
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lemma restrict_apply [simp]:
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    "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
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  by (simp add: restrict_def)
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lemma restrict_ext:
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    "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
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  by (simp add: 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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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_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 inv_into_funcset: "f ` A = B ==> (\<lambda>x\<in>B. inv_into A f x) : B -> A"
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by (unfold inv_into_def) (fast intro: someI2)
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lemma compose_inv_into_id:
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  "bij_betw f A B ==> compose A (\<lambda>y\<in>B. inv_into A f y) f = (\<lambda>x\<in>A. x)"
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apply (simp add: bij_betw_def compose_def)
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apply (rule restrict_ext, auto)
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done
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lemma compose_id_inv_into:
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  "f ` A = B ==> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
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apply (simp add: compose_def)
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apply (rule restrict_ext)
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apply (simp add: f_inv_into_f)
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done
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subsection{*Cardinality*}
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lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
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by (rule card_inj_on_le) auto
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lemma card_bij:
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  "[|f \<in> A\<rightarrow>B; inj_on f A;
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     g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
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by (blast intro: card_inj order_antisym)
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subsection {* Extensional Function Spaces *} 
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definition extensional_funcset
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where "extensional_funcset S T = (S -> T) \<inter> (extensional S)"
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lemma extensional_empty[simp]: "extensional {} = {%x. undefined}"
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unfolding extensional_def by auto
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lemma extensional_funcset_empty_domain: "extensional_funcset {} T = {%x. undefined}"
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unfolding extensional_funcset_def by simp
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lemma extensional_funcset_empty_range:
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  assumes "S \<noteq> {}"
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  shows "extensional_funcset S {} = {}"
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using assms unfolding extensional_funcset_def by auto
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lemma extensional_funcset_arb:
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  assumes "f \<in> extensional_funcset S T" "x \<notin> S"
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  shows "f x = undefined"
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using assms
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unfolding extensional_funcset_def by auto (auto dest!: extensional_arb)
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lemma extensional_funcset_mem: "f \<in> extensional_funcset S T \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<in> T"
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unfolding extensional_funcset_def by auto
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lemma extensional_subset: "f : extensional A ==> A <= B ==> f : extensional B"
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unfolding extensional_def by auto
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lemma extensional_funcset_extend_domainI: "\<lbrakk> y \<in> T; f \<in> extensional_funcset S T\<rbrakk> \<Longrightarrow> f(x := y) \<in> extensional_funcset (insert x S) T"
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unfolding extensional_funcset_def extensional_def by auto
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lemma extensional_funcset_restrict_domain:
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  "x \<notin> S \<Longrightarrow> f \<in> extensional_funcset (insert x S) T \<Longrightarrow> f(x := undefined) \<in> extensional_funcset S T"
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unfolding extensional_funcset_def extensional_def by auto
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lemma extensional_funcset_extend_domain_eq:
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  assumes "x \<notin> S"
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  shows
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    "extensional_funcset (insert x S) T = (\<lambda>(y, g). g(x := y)) ` {(y, g). y \<in> T \<and> g \<in> extensional_funcset S T}"
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  using assms
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proof -
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  {
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    fix f
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    assume "f : extensional_funcset (insert x S) T"
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    from this assms have "f : (%(y, g). g(x := y)) ` (T <*> extensional_funcset S T)"
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      unfolding image_iff
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      apply (rule_tac x="(f x, f(x := undefined))" in bexI)
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    apply (auto intro: extensional_funcset_extend_domainI extensional_funcset_restrict_domain extensional_funcset_mem) done 
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  }
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  moreover
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  {
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    fix f
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    assume "f : (%(y, g). g(x := y)) ` (T <*> extensional_funcset S T)"
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    from this assms have "f : extensional_funcset (insert x S) T"
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      by (auto intro: extensional_funcset_extend_domainI)
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  }
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  ultimately show ?thesis by auto
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qed
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lemma extensional_funcset_fun_upd_restricts_rangeI:  "\<forall> y \<in> S. f x \<noteq> f y ==> f : extensional_funcset (insert x S) T ==> f(x := undefined) : extensional_funcset S (T - {f x})"
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unfolding extensional_funcset_def extensional_def
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apply auto
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apply (case_tac "x = xa")
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apply auto done
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lemma extensional_funcset_fun_upd_extends_rangeI:
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  assumes "a \<in> T" "f : extensional_funcset S (T - {a})"
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  shows "f(x := a) : extensional_funcset (insert x S) T"
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  using assms unfolding extensional_funcset_def extensional_def by auto
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subsubsection {* Injective Extensional Function Spaces *}
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lemma extensional_funcset_fun_upd_inj_onI:
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  assumes "f \<in> extensional_funcset S (T - {a})" "inj_on f S"
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  shows "inj_on (f(x := a)) S"
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  using assms unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
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lemma extensional_funcset_extend_domain_inj_on_eq:
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  assumes "x \<notin> S"
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  shows"{f. f \<in> extensional_funcset (insert x S) T \<and> inj_on f (insert x S)} =
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    (%(y, g). g(x:=y)) ` {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S}"
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proof -
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  from assms show ?thesis
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    apply auto
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    apply (auto intro: extensional_funcset_fun_upd_inj_onI extensional_funcset_fun_upd_extends_rangeI)
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    apply (auto simp add: image_iff inj_on_def)
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    apply (rule_tac x="xa x" in exI)
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    apply (auto intro: extensional_funcset_mem)
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    apply (rule_tac x="xa(x := undefined)" in exI)
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    apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
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    apply (auto dest!: extensional_funcset_mem split: split_if_asm)
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    done
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qed
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lemma extensional_funcset_extend_domain_inj_onI:
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  assumes "x \<notin> S"
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  shows "inj_on (\<lambda>(y, g). g(x := y)) {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S}"
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proof -
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  from assms show ?thesis
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    apply (auto intro!: inj_onI)
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    apply (metis fun_upd_same)
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    by (metis assms extensional_funcset_arb fun_upd_triv fun_upd_upd)
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qed
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subsubsection {* Cardinality *}
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lemma card_extensional_funcset:
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  assumes "finite S"
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  shows "card (extensional_funcset S T) = (card T) ^ (card S)"
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using assms
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proof (induct rule: finite_induct)
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  case empty
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  show ?case
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    by (auto simp add: extensional_funcset_empty_domain)
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next
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  case (insert x S)
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  {
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    fix g g' y y'
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    assume assms: "g \<in> extensional_funcset S T"
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      "g' \<in> extensional_funcset S T"
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      "y \<in> T" "y' \<in> T"
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      "g(x := y) = g'(x := y')"
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    from this have "y = y'"
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      by (metis fun_upd_same)
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    have "g = g'"
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      by (metis assms(1) assms(2) assms(5) extensional_funcset_arb fun_upd_triv fun_upd_upd insert(2))
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  from `y = y'` `g = g'` have "y = y' & g = g'" by simp
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  }
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  from this have "inj_on (\<lambda>(y, g). g (x := y)) (T \<times> extensional_funcset S T)"
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   390
    by (auto intro: inj_onI)
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  from this insert.hyps show ?case
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   392
    by (simp add: extensional_funcset_extend_domain_eq card_image card_cartesian_product)
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qed
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   394
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   395
lemma finite_extensional_funcset:
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  assumes "finite S" "finite T"
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   397
  shows "finite (extensional_funcset S T)"
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   398
proof -
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   399
  from card_extensional_funcset[OF assms(1), of T] assms(2)
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   400
  have "(card (extensional_funcset S T) \<noteq> 0) \<or> (S \<noteq> {} \<and> T = {})"
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   401
    by auto
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   402
  from this show ?thesis
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   403
  proof
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   404
    assume "card (extensional_funcset S T) \<noteq> 0"
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   405
    from this show ?thesis
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   406
      by (auto intro: card_ge_0_finite)
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   407
  next
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   408
    assume "S \<noteq> {} \<and> T = {}"
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   409
    from this show ?thesis
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   410
      by (auto simp add: extensional_funcset_empty_range)
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   411
  qed
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   412
qed
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   413
paulson@13586
   414
end