src/HOL/Library/FuncSet.thy
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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 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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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 (simp add: Pi_def, blast)
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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 (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