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(* Title: HOL/Library/FuncSet.thy


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ID: $Id$


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Author: Florian Kammueller and Lawrence C Paulson


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*)


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header {* Pi and Function Sets *}

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theory FuncSet = Main:


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constdefs

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Pi :: "['a set, 'a => 'b set] => ('a => 'b) set"


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"Pi A B == {f. \<forall>x. x \<in> A > f x \<in> B x}"

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extensional :: "'a set => ('a => 'b) set"

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"extensional A == {f. \<forall>x. x~:A > f x = arbitrary}"

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"restrict" :: "['a => 'b, 'a set] => ('a => 'b)"


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"restrict f A == (%x. if x \<in> A then f x else arbitrary)"

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syntax


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"@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3PI _:_./ _)" 10)


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funcset :: "['a set, 'b set] => ('a => 'b) set" (infixr ">" 60)


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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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funcset :: "['a set, 'b set] => ('a => 'b) set" (infixr "\<rightarrow>" 60)

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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" => "Pi A (%x. B)"

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"A > B" => "Pi A (_K B)"


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"%x:A. f" == "restrict (%x. f) A"

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constdefs

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"compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"

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"compose A g f == \<lambda>x\<in>A. g (f x)"


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print_translation {* [("Pi", dependent_tr' ("@Pi", "funcset"))] *}

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subsection{*Basic Properties of @{term Pi}*}


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

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by (simp add: Pi_def)

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

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by (simp add: Pi_def)

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lemma Pi_mem: "[f: Pi A B; x \<in> A] ==> f x \<in> B x"

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by (simp add: Pi_def)

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lemma funcset_mem: "[f \<in> A > B; x \<in> A] ==> f x \<in> B"

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by (simp add: Pi_def)

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lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"


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by (auto simp add: Pi_def)


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lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"

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apply (simp add: Pi_def, auto)

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txt{*Converse direction requires Axiom of Choice to exhibit a function


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picking an element from each nonempty @{term "B x"}*}

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

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

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done


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lemma Pi_empty [simp]: "Pi {} B = UNIV"

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by (simp add: Pi_def)

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lemma Pi_UNIV [simp]: "A > UNIV = UNIV"

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by (simp add: Pi_def)

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text{*Covariance of Pisets in their second argument*}


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lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"

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by (simp add: Pi_def, blast)

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text{*Contravariance of Pisets in their first argument*}


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

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by (simp add: Pi_def, blast)

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subsection{*Composition With a Restricted Domain: @{term compose}*}


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lemma funcset_compose:


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


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by (simp add: Pi_def compose_def restrict_def)

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lemma compose_assoc:

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"[ f \<in> A > B; g \<in> B > C; h \<in> C > D ]

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

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by (simp add: expand_fun_eq Pi_def compose_def restrict_def)

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lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"

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by (simp add: compose_def restrict_def)

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lemma surj_compose: "[ f ` A = B; g ` B = C ] ==> compose A g f ` A = C"

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by (auto simp add: image_def compose_eq)

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lemma inj_on_compose:

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"[ f ` A = B; inj_on f A; inj_on g B ] ==> inj_on (compose A g f) A"


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by (auto simp add: inj_on_def compose_eq)

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subsection{*Bounded Abstraction: @{term restrict}*}


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

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by (simp add: Pi_def restrict_def)

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

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by (simp add: Pi_def restrict_def)

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lemma restrict_apply [simp]:

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"(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else arbitrary)"


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by (simp add: restrict_def)

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lemma restrict_ext:

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"(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"

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by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)

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lemma inj_on_restrict_eq: "inj_on (restrict f A) A = inj_on f A"

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by (simp add: inj_on_def restrict_def)

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lemma Id_compose:

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"[f \<in> A > B; f \<in> extensional A] ==> compose A (\<lambda>y\<in>B. y) f = f"


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by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)

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lemma compose_Id:

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"[g \<in> A > B; g \<in> extensional A] ==> compose A g (\<lambda>x\<in>A. x) = g"


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by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)

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subsection{*Extensionality*}


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lemma extensional_arb: "[f \<in> extensional A; x\<notin> A] ==> f x = arbitrary"

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by (simp add: extensional_def)

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

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by (simp add: restrict_def extensional_def)

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

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by (simp add: compose_def)

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lemma extensionalityI:

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


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


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by (force simp add: expand_fun_eq extensional_def)

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

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

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lemma compose_Inv_id:

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"[ inj_on f A; f ` A = B ]

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

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apply (simp add: compose_def)


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apply (rule restrict_ext, auto)


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apply (erule subst)


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apply (simp add: Inv_f_f)


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done

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lemma compose_id_Inv:

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


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apply (simp add: compose_def)


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apply (rule restrict_ext)


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apply (simp add: f_Inv_f)


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done

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subsection{*Bijections Between Sets*}


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text{*The basic definition could be moved to @{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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constdefs


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bij_betw :: "['a => 'b, 'a set, 'b set] => bool" (*bijective*)


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"bij_betw f A B == inj_on f A & f ` A = B"


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lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"


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by (simp add: bij_betw_def)


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lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"


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by (auto simp add: bij_betw_def inj_on_Inv Pi_def)


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lemma bij_betw_Inv: "bij_betw f A B \<Longrightarrow> bij_betw (Inv A f) B A"


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apply (auto simp add: bij_betw_def inj_on_Inv Inv_mem)


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apply (simp add: image_compose [symmetric] o_def)


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apply (simp add: image_def Inv_f_f)


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done


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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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subsection{*Cardinality*}


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


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apply (rule card_inj_on_le)


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apply (auto simp add: Pi_def)


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done


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lemma card_bij:


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


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


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by (blast intro: card_inj order_antisym)


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end
