src/HOL/Hilbert_Choice.thy
author hoelzl
Fri Mar 22 10:41:43 2013 +0100 (2013-03-22)
changeset 51474 1e9e68247ad1
parent 50105 65d5b18e1626
child 52143 36ffe23b25f8
permissions -rw-r--r--
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
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(*  Title:      HOL/Hilbert_Choice.thy
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    Author:     Lawrence C Paulson, Tobias Nipkow
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    Copyright   2001  University of Cambridge
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*)
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header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
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theory Hilbert_Choice
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imports Nat Wellfounded Big_Operators
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keywords "specification" "ax_specification" :: thy_goal
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begin
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subsection {* Hilbert's epsilon *}
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axiomatization Eps :: "('a => bool) => 'a" where
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  someI: "P x ==> P (Eps P)"
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syntax (epsilon)
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  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
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syntax (HOL)
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  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
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syntax
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  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
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translations
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  "SOME x. P" == "CONST Eps (%x. P)"
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print_translation {*
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  [(@{const_syntax Eps}, fn [Abs abs] =>
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      let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
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      in Syntax.const @{syntax_const "_Eps"} $ x $ t end)]
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*} -- {* to avoid eta-contraction of body *}
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definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
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"inv_into A f == %x. SOME y. y : A & f y = x"
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abbreviation inv :: "('a => 'b) => ('b => 'a)" where
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"inv == inv_into UNIV"
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subsection {*Hilbert's Epsilon-operator*}
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text{*Easier to apply than @{text someI} if the witness comes from an
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existential formula*}
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lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
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apply (erule exE)
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apply (erule someI)
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done
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text{*Easier to apply than @{text someI} because the conclusion has only one
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occurrence of @{term P}.*}
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lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
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by (blast intro: someI)
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text{*Easier to apply than @{text someI2} if the witness comes from an
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existential formula*}
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lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
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by (blast intro: someI2)
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lemma some_equality [intro]:
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     "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
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by (blast intro: someI2)
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lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
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by blast
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lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
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by (blast intro: someI)
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lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
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apply (rule some_equality)
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apply (rule refl, assumption)
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done
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lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
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apply (rule some_equality)
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apply (rule refl)
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apply (erule sym)
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done
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subsection{*Axiom of Choice, Proved Using the Description Operator*}
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lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
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by (fast elim: someI)
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lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
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by (fast elim: someI)
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lemma choice_iff: "(\<forall>x. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x. Q x (f x))"
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by (fast elim: someI)
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lemma choice_iff': "(\<forall>x. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x. P x \<longrightarrow> Q x (f x))"
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by (fast elim: someI)
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lemma bchoice_iff: "(\<forall>x\<in>S. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. Q x (f x))"
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by (fast elim: someI)
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lemma bchoice_iff': "(\<forall>x\<in>S. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. P x \<longrightarrow> Q x (f x))"
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by (fast elim: someI)
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subsection {*Function Inverse*}
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lemma inv_def: "inv f = (%y. SOME x. f x = y)"
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by(simp add: inv_into_def)
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lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
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apply (simp add: inv_into_def)
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apply (fast intro: someI2)
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done
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lemma inv_id [simp]: "inv id = id"
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by (simp add: inv_into_def id_def)
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lemma inv_into_f_f [simp]:
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  "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
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apply (simp add: inv_into_def inj_on_def)
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apply (blast intro: someI2)
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done
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lemma inv_f_f: "inj f ==> inv f (f x) = x"
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by simp
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lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
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apply (simp add: inv_into_def)
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apply (fast intro: someI2)
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done
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lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
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apply (erule subst)
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apply (fast intro: inv_into_f_f)
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done
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lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
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by (simp add:inv_into_f_eq)
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lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
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  by (blast intro: inv_into_f_eq)
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text{*But is it useful?*}
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lemma inj_transfer:
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  assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
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  shows "P x"
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proof -
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  have "f x \<in> range f" by auto
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  hence "P(inv f (f x))" by (rule minor)
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  thus "P x" by (simp add: inv_into_f_f [OF injf])
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qed
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lemma inj_iff: "(inj f) = (inv f o f = id)"
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apply (simp add: o_def fun_eq_iff)
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apply (blast intro: inj_on_inverseI inv_into_f_f)
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done
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lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
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by (simp add: inj_iff)
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lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
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by (simp add: comp_assoc)
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lemma inv_into_image_cancel[simp]:
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  "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
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by(fastforce simp: image_def)
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lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
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by (blast intro!: surjI inv_into_f_f)
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lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
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by (simp add: f_inv_into_f)
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lemma inv_into_injective:
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  assumes eq: "inv_into A f x = inv_into A f y"
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      and x: "x: f`A"
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      and y: "y: f`A"
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  shows "x=y"
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proof -
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  have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
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  thus ?thesis by (simp add: f_inv_into_f x y)
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qed
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lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
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by (blast intro: inj_onI dest: inv_into_injective injD)
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lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
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by (auto simp add: bij_betw_def inj_on_inv_into)
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lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
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by (simp add: inj_on_inv_into)
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lemma surj_iff: "(surj f) = (f o inv f = id)"
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by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
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lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
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  unfolding surj_iff by (simp add: o_def fun_eq_iff)
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lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
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apply (rule ext)
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apply (drule_tac x = "inv f x" in spec)
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apply (simp add: surj_f_inv_f)
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done
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lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
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by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
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lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
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apply (rule ext)
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apply (auto simp add: inv_into_def)
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done
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lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
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apply (rule inv_equality)
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apply (auto simp add: bij_def surj_f_inv_f)
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done
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(** bij(inv f) implies little about f.  Consider f::bool=>bool such that
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    f(True)=f(False)=True.  Then it's consistent with axiom someI that
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    inv f could be any function at all, including the identity function.
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    If inv f=id then inv f is a bijection, but inj f, surj(f) and
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    inv(inv f)=f all fail.
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**)
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lemma inv_into_comp:
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  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
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  inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
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apply (rule inv_into_f_eq)
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  apply (fast intro: comp_inj_on)
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 apply (simp add: inv_into_into)
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apply (simp add: f_inv_into_f inv_into_into)
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done
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lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
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apply (rule inv_equality)
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apply (auto simp add: bij_def surj_f_inv_f)
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done
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lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
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by (simp add: image_eq_UN surj_f_inv_f)
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lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
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by (simp add: image_eq_UN)
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lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
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by (auto simp add: image_def)
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lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
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apply auto
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apply (force simp add: bij_is_inj)
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apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
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done
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lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A" 
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apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
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apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
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done
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lemma finite_fun_UNIVD1:
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  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
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  and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
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  shows "finite (UNIV :: 'a set)"
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proof -
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  from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
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  with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
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    by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
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  then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
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  then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
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  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
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  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
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  proof (rule UNIV_eq_I)
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    fix x :: 'a
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    from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
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    thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
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  qed
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  ultimately show "finite (UNIV :: 'a set)" by simp
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qed
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lemma image_inv_into_cancel:
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  assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
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  shows "f `((inv_into A f)`B') = B'"
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  using assms
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proof (auto simp add: f_inv_into_f)
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  let ?f' = "(inv_into A f)"
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  fix a' assume *: "a' \<in> B'"
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  then have "a' \<in> A'" using SUB by auto
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  then have "a' = f (?f' a')"
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    using SURJ by (auto simp add: f_inv_into_f)
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  then show "a' \<in> f ` (?f' ` B')" using * by blast
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qed
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lemma inv_into_inv_into_eq:
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  assumes "bij_betw f A A'" "a \<in> A"
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  shows "inv_into A' (inv_into A f) a = f a"
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proof -
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  let ?f' = "inv_into A f"   let ?f'' = "inv_into A' ?f'"
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  have 1: "bij_betw ?f' A' A" using assms
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  by (auto simp add: bij_betw_inv_into)
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  obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
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    using 1 `a \<in> A` unfolding bij_betw_def by force
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  hence "?f'' a = a'"
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    using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
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  moreover have "f a = a'" using assms 2 3
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    by (auto simp add: bij_betw_def)
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  ultimately show "?f'' a = f a" by simp
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qed
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lemma inj_on_iff_surj:
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  assumes "A \<noteq> {}"
hoelzl@40703
   306
  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
hoelzl@40703
   307
proof safe
hoelzl@40703
   308
  fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
hoelzl@40703
   309
  let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"  let ?csi = "\<lambda>a. a \<in> A"
hoelzl@40703
   310
  let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
hoelzl@40703
   311
  have "?g ` A' = A"
hoelzl@40703
   312
  proof
hoelzl@40703
   313
    show "?g ` A' \<le> A"
hoelzl@40703
   314
    proof clarify
hoelzl@40703
   315
      fix a' assume *: "a' \<in> A'"
hoelzl@40703
   316
      show "?g a' \<in> A"
hoelzl@40703
   317
      proof cases
hoelzl@40703
   318
        assume Case1: "a' \<in> f ` A"
hoelzl@40703
   319
        then obtain a where "?phi a' a" by blast
hoelzl@40703
   320
        hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
hoelzl@40703
   321
        with Case1 show ?thesis by auto
hoelzl@40703
   322
      next
hoelzl@40703
   323
        assume Case2: "a' \<notin> f ` A"
hoelzl@40703
   324
        hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
hoelzl@40703
   325
        with Case2 show ?thesis by auto
hoelzl@40703
   326
      qed
hoelzl@40703
   327
    qed
hoelzl@40703
   328
  next
hoelzl@40703
   329
    show "A \<le> ?g ` A'"
hoelzl@40703
   330
    proof-
hoelzl@40703
   331
      {fix a assume *: "a \<in> A"
hoelzl@40703
   332
       let ?b = "SOME aa. ?phi (f a) aa"
hoelzl@40703
   333
       have "?phi (f a) a" using * by auto
hoelzl@40703
   334
       hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
hoelzl@40703
   335
       hence "?g(f a) = ?b" using * by auto
hoelzl@40703
   336
       moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
hoelzl@40703
   337
       ultimately have "?g(f a) = a" by simp
hoelzl@40703
   338
       with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
hoelzl@40703
   339
      }
hoelzl@40703
   340
      thus ?thesis by force
hoelzl@40703
   341
    qed
hoelzl@40703
   342
  qed
hoelzl@40703
   343
  thus "\<exists>g. g ` A' = A" by blast
hoelzl@40703
   344
next
hoelzl@40703
   345
  fix g  let ?f = "inv_into A' g"
hoelzl@40703
   346
  have "inj_on ?f (g ` A')"
hoelzl@40703
   347
    by (auto simp add: inj_on_inv_into)
hoelzl@40703
   348
  moreover
hoelzl@40703
   349
  {fix a' assume *: "a' \<in> A'"
hoelzl@40703
   350
   let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
hoelzl@40703
   351
   have "?phi a'" using * by auto
hoelzl@40703
   352
   hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
hoelzl@40703
   353
   hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
hoelzl@40703
   354
  }
hoelzl@40703
   355
  ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
hoelzl@40703
   356
qed
hoelzl@40703
   357
hoelzl@40703
   358
lemma Ex_inj_on_UNION_Sigma:
hoelzl@40703
   359
  "\<exists>f. (inj_on f (\<Union> i \<in> I. A i) \<and> f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i))"
hoelzl@40703
   360
proof
hoelzl@40703
   361
  let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
hoelzl@40703
   362
  let ?sm = "\<lambda> a. SOME i. ?phi a i"
hoelzl@40703
   363
  let ?f = "\<lambda>a. (?sm a, a)"
hoelzl@40703
   364
  have "inj_on ?f (\<Union> i \<in> I. A i)" unfolding inj_on_def by auto
hoelzl@40703
   365
  moreover
hoelzl@40703
   366
  { { fix i a assume "i \<in> I" and "a \<in> A i"
hoelzl@40703
   367
      hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
hoelzl@40703
   368
    }
hoelzl@40703
   369
    hence "?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
hoelzl@40703
   370
  }
hoelzl@40703
   371
  ultimately
hoelzl@40703
   372
  show "inj_on ?f (\<Union> i \<in> I. A i) \<and> ?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)"
hoelzl@40703
   373
  by auto
hoelzl@40703
   374
qed
hoelzl@40703
   375
hoelzl@40703
   376
subsection {* The Cantor-Bernstein Theorem *}
hoelzl@40703
   377
hoelzl@40703
   378
lemma Cantor_Bernstein_aux:
hoelzl@40703
   379
  shows "\<exists>A' h. A' \<le> A \<and>
hoelzl@40703
   380
                (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
hoelzl@40703
   381
                (\<forall>a \<in> A'. h a = f a) \<and>
hoelzl@40703
   382
                (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
hoelzl@40703
   383
proof-
hoelzl@40703
   384
  obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
hoelzl@40703
   385
  have 0: "mono H" unfolding mono_def H_def by blast
hoelzl@40703
   386
  then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
hoelzl@40703
   387
  hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
hoelzl@40703
   388
  hence 3: "A' \<le> A" by blast
hoelzl@40703
   389
  have 4: "\<forall>a \<in> A'.  a \<notin> g`(B - f ` A')"
hoelzl@40703
   390
  using 2 by blast
hoelzl@40703
   391
  have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
hoelzl@40703
   392
  using 2 by blast
hoelzl@40703
   393
  (*  *)
hoelzl@40703
   394
  obtain h where h_def:
hoelzl@40703
   395
  "h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
hoelzl@40703
   396
  hence "\<forall>a \<in> A'. h a = f a" by auto
hoelzl@40703
   397
  moreover
hoelzl@40703
   398
  have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
hoelzl@40703
   399
  proof
hoelzl@40703
   400
    fix a assume *: "a \<in> A - A'"
hoelzl@40703
   401
    let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
hoelzl@40703
   402
    have "h a = (SOME b. ?phi b)" using h_def * by auto
hoelzl@40703
   403
    moreover have "\<exists>b. ?phi b" using 5 *  by auto
hoelzl@40703
   404
    ultimately show  "?phi (h a)" using someI_ex[of ?phi] by auto
hoelzl@40703
   405
  qed
hoelzl@40703
   406
  ultimately show ?thesis using 3 4 by blast
hoelzl@40703
   407
qed
hoelzl@40703
   408
hoelzl@40703
   409
theorem Cantor_Bernstein:
hoelzl@40703
   410
  assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
hoelzl@40703
   411
          INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
hoelzl@40703
   412
  shows "\<exists>h. bij_betw h A B"
hoelzl@40703
   413
proof-
hoelzl@40703
   414
  obtain A' and h where 0: "A' \<le> A" and
hoelzl@40703
   415
  1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
hoelzl@40703
   416
  2: "\<forall>a \<in> A'. h a = f a" and
hoelzl@40703
   417
  3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
hoelzl@40703
   418
  using Cantor_Bernstein_aux[of A g B f] by blast
hoelzl@40703
   419
  have "inj_on h A"
hoelzl@40703
   420
  proof (intro inj_onI)
hoelzl@40703
   421
    fix a1 a2
hoelzl@40703
   422
    assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
hoelzl@40703
   423
    show "a1 = a2"
hoelzl@40703
   424
    proof(cases "a1 \<in> A'")
hoelzl@40703
   425
      assume Case1: "a1 \<in> A'"
hoelzl@40703
   426
      show ?thesis
hoelzl@40703
   427
      proof(cases "a2 \<in> A'")
hoelzl@40703
   428
        assume Case11: "a2 \<in> A'"
hoelzl@40703
   429
        hence "f a1 = f a2" using Case1 2 6 by auto
hoelzl@40703
   430
        thus ?thesis using INJ1 Case1 Case11 0
hoelzl@40703
   431
        unfolding inj_on_def by blast
hoelzl@40703
   432
      next
hoelzl@40703
   433
        assume Case12: "a2 \<notin> A'"
hoelzl@40703
   434
        hence False using 3 5 2 6 Case1 by force
hoelzl@40703
   435
        thus ?thesis by simp
hoelzl@40703
   436
      qed
hoelzl@40703
   437
    next
hoelzl@40703
   438
    assume Case2: "a1 \<notin> A'"
hoelzl@40703
   439
      show ?thesis
hoelzl@40703
   440
      proof(cases "a2 \<in> A'")
hoelzl@40703
   441
        assume Case21: "a2 \<in> A'"
hoelzl@40703
   442
        hence False using 3 4 2 6 Case2 by auto
hoelzl@40703
   443
        thus ?thesis by simp
hoelzl@40703
   444
      next
hoelzl@40703
   445
        assume Case22: "a2 \<notin> A'"
hoelzl@40703
   446
        hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
hoelzl@40703
   447
        thus ?thesis using 6 by simp
hoelzl@40703
   448
      qed
hoelzl@40703
   449
    qed
hoelzl@40703
   450
  qed
hoelzl@40703
   451
  (*  *)
hoelzl@40703
   452
  moreover
hoelzl@40703
   453
  have "h ` A = B"
hoelzl@40703
   454
  proof safe
hoelzl@40703
   455
    fix a assume "a \<in> A"
wenzelm@47988
   456
    thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto
hoelzl@40703
   457
  next
hoelzl@40703
   458
    fix b assume *: "b \<in> B"
hoelzl@40703
   459
    show "b \<in> h ` A"
hoelzl@40703
   460
    proof(cases "b \<in> f ` A'")
hoelzl@40703
   461
      assume Case1: "b \<in> f ` A'"
hoelzl@40703
   462
      then obtain a where "a \<in> A' \<and> b = f a" by blast
hoelzl@40703
   463
      thus ?thesis using 2 0 by force
hoelzl@40703
   464
    next
hoelzl@40703
   465
      assume Case2: "b \<notin> f ` A'"
hoelzl@40703
   466
      hence "g b \<notin> A'" using 1 * by auto
hoelzl@40703
   467
      hence 4: "g b \<in> A - A'" using * SUB2 by auto
hoelzl@40703
   468
      hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
hoelzl@40703
   469
      using 3 by auto
hoelzl@40703
   470
      hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
hoelzl@40703
   471
      thus ?thesis using 4 by force
hoelzl@40703
   472
    qed
hoelzl@40703
   473
  qed
hoelzl@40703
   474
  (*  *)
hoelzl@40703
   475
  ultimately show ?thesis unfolding bij_betw_def by auto
hoelzl@40703
   476
qed
paulson@14760
   477
paulson@14760
   478
subsection {*Other Consequences of Hilbert's Epsilon*}
paulson@14760
   479
paulson@14760
   480
text {*Hilbert's Epsilon and the @{term split} Operator*}
paulson@14760
   481
paulson@14760
   482
text{*Looping simprule*}
paulson@14760
   483
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
haftmann@26347
   484
  by simp
paulson@14760
   485
paulson@14760
   486
lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
haftmann@26347
   487
  by (simp add: split_def)
paulson@14760
   488
paulson@14760
   489
lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
haftmann@26347
   490
  by blast
paulson@14760
   491
paulson@14760
   492
paulson@14760
   493
text{*A relation is wellfounded iff it has no infinite descending chain*}
paulson@14760
   494
lemma wf_iff_no_infinite_down_chain:
paulson@14760
   495
  "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
paulson@14760
   496
apply (simp only: wf_eq_minimal)
paulson@14760
   497
apply (rule iffI)
paulson@14760
   498
 apply (rule notI)
paulson@14760
   499
 apply (erule exE)
paulson@14760
   500
 apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
paulson@14760
   501
apply (erule contrapos_np, simp, clarify)
paulson@14760
   502
apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
paulson@14760
   503
 apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
paulson@14760
   504
 apply (rule allI, simp)
paulson@14760
   505
 apply (rule someI2_ex, blast, blast)
paulson@14760
   506
apply (rule allI)
paulson@14760
   507
apply (induct_tac "n", simp_all)
paulson@14760
   508
apply (rule someI2_ex, blast+)
paulson@14760
   509
done
paulson@14760
   510
nipkow@27760
   511
lemma wf_no_infinite_down_chainE:
nipkow@27760
   512
  assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
nipkow@27760
   513
using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
nipkow@27760
   514
nipkow@27760
   515
paulson@14760
   516
text{*A dynamically-scoped fact for TFL *}
wenzelm@12298
   517
lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
wenzelm@12298
   518
  by (blast intro: someI)
paulson@11451
   519
wenzelm@12298
   520
wenzelm@12298
   521
subsection {* Least value operator *}
paulson@11451
   522
haftmann@35416
   523
definition
haftmann@35416
   524
  LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
paulson@14760
   525
  "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
paulson@11451
   526
paulson@11451
   527
syntax
wenzelm@12298
   528
  "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   529
translations
wenzelm@35115
   530
  "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
paulson@11451
   531
paulson@11451
   532
lemma LeastMI2:
wenzelm@12298
   533
  "P x ==> (!!y. P y ==> m x <= m y)
wenzelm@12298
   534
    ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
wenzelm@12298
   535
    ==> Q (LeastM m P)"
paulson@14760
   536
  apply (simp add: LeastM_def)
paulson@14208
   537
  apply (rule someI2_ex, blast, blast)
wenzelm@12298
   538
  done
paulson@11451
   539
paulson@11451
   540
lemma LeastM_equality:
wenzelm@12298
   541
  "P k ==> (!!x. P x ==> m k <= m x)
wenzelm@12298
   542
    ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
paulson@14208
   543
  apply (rule LeastMI2, assumption, blast)
wenzelm@12298
   544
  apply (blast intro!: order_antisym)
wenzelm@12298
   545
  done
paulson@11451
   546
paulson@11454
   547
lemma wf_linord_ex_has_least:
paulson@14760
   548
  "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
paulson@14760
   549
    ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
wenzelm@12298
   550
  apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
paulson@14208
   551
  apply (drule_tac x = "m`Collect P" in spec, force)
wenzelm@12298
   552
  done
paulson@11454
   553
paulson@11454
   554
lemma ex_has_least_nat:
paulson@14760
   555
    "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
wenzelm@12298
   556
  apply (simp only: pred_nat_trancl_eq_le [symmetric])
wenzelm@12298
   557
  apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
paulson@16796
   558
   apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
wenzelm@12298
   559
  done
paulson@11454
   560
wenzelm@12298
   561
lemma LeastM_nat_lemma:
paulson@14760
   562
    "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
paulson@14760
   563
  apply (simp add: LeastM_def)
wenzelm@12298
   564
  apply (rule someI_ex)
wenzelm@12298
   565
  apply (erule ex_has_least_nat)
wenzelm@12298
   566
  done
paulson@11454
   567
wenzelm@45607
   568
lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1]
paulson@11454
   569
paulson@11454
   570
lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
paulson@14208
   571
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
paulson@11454
   572
paulson@11451
   573
wenzelm@12298
   574
subsection {* Greatest value operator *}
paulson@11451
   575
haftmann@35416
   576
definition
haftmann@35416
   577
  GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
paulson@14760
   578
  "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
wenzelm@12298
   579
haftmann@35416
   580
definition
haftmann@35416
   581
  Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
wenzelm@12298
   582
  "Greatest == GreatestM (%x. x)"
paulson@11451
   583
paulson@11451
   584
syntax
wenzelm@35115
   585
  "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
wenzelm@12298
   586
      ("GREATEST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   587
translations
wenzelm@35115
   588
  "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
paulson@11451
   589
paulson@11451
   590
lemma GreatestMI2:
wenzelm@12298
   591
  "P x ==> (!!y. P y ==> m y <= m x)
wenzelm@12298
   592
    ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
wenzelm@12298
   593
    ==> Q (GreatestM m P)"
paulson@14760
   594
  apply (simp add: GreatestM_def)
paulson@14208
   595
  apply (rule someI2_ex, blast, blast)
wenzelm@12298
   596
  done
paulson@11451
   597
paulson@11451
   598
lemma GreatestM_equality:
wenzelm@12298
   599
 "P k ==> (!!x. P x ==> m x <= m k)
wenzelm@12298
   600
    ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
paulson@14208
   601
  apply (rule_tac m = m in GreatestMI2, assumption, blast)
wenzelm@12298
   602
  apply (blast intro!: order_antisym)
wenzelm@12298
   603
  done
paulson@11451
   604
paulson@11451
   605
lemma Greatest_equality:
wenzelm@12298
   606
  "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
paulson@14760
   607
  apply (simp add: Greatest_def)
paulson@14208
   608
  apply (erule GreatestM_equality, blast)
wenzelm@12298
   609
  done
paulson@11451
   610
paulson@11451
   611
lemma ex_has_greatest_nat_lemma:
paulson@14760
   612
  "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
paulson@14760
   613
    ==> \<exists>y. P y & ~ (m y < m k + n)"
paulson@15251
   614
  apply (induct n, force)
wenzelm@12298
   615
  apply (force simp add: le_Suc_eq)
wenzelm@12298
   616
  done
paulson@11451
   617
wenzelm@12298
   618
lemma ex_has_greatest_nat:
paulson@14760
   619
  "P k ==> \<forall>y. P y --> m y < b
paulson@14760
   620
    ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
wenzelm@12298
   621
  apply (rule ccontr)
wenzelm@12298
   622
  apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
paulson@14208
   623
    apply (subgoal_tac [3] "m k <= b", auto)
wenzelm@12298
   624
  done
paulson@11451
   625
wenzelm@12298
   626
lemma GreatestM_nat_lemma:
paulson@14760
   627
  "P k ==> \<forall>y. P y --> m y < b
paulson@14760
   628
    ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
paulson@14760
   629
  apply (simp add: GreatestM_def)
wenzelm@12298
   630
  apply (rule someI_ex)
paulson@14208
   631
  apply (erule ex_has_greatest_nat, assumption)
wenzelm@12298
   632
  done
paulson@11451
   633
wenzelm@45607
   634
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]
paulson@11451
   635
wenzelm@12298
   636
lemma GreatestM_nat_le:
paulson@14760
   637
  "P x ==> \<forall>y. P y --> m y < b
wenzelm@12298
   638
    ==> (m x::nat) <= m (GreatestM m P)"
berghofe@21020
   639
  apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
wenzelm@12298
   640
  done
wenzelm@12298
   641
wenzelm@12298
   642
wenzelm@12298
   643
text {* \medskip Specialization to @{text GREATEST}. *}
wenzelm@12298
   644
paulson@14760
   645
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
paulson@14760
   646
  apply (simp add: Greatest_def)
paulson@14208
   647
  apply (rule GreatestM_natI, auto)
wenzelm@12298
   648
  done
paulson@11451
   649
wenzelm@12298
   650
lemma Greatest_le:
paulson@14760
   651
    "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
paulson@14760
   652
  apply (simp add: Greatest_def)
paulson@14208
   653
  apply (rule GreatestM_nat_le, auto)
wenzelm@12298
   654
  done
wenzelm@12298
   655
wenzelm@12298
   656
haftmann@49948
   657
subsection {* An aside: bounded accessible part *}
haftmann@49948
   658
haftmann@49948
   659
text {* Finite monotone eventually stable sequences *}
haftmann@49948
   660
haftmann@49948
   661
lemma finite_mono_remains_stable_implies_strict_prefix:
haftmann@49948
   662
  fixes f :: "nat \<Rightarrow> 'a::order"
haftmann@49948
   663
  assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
haftmann@49948
   664
  shows "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m < f n) \<and> (\<forall>n\<ge>N. f N = f n)"
haftmann@49948
   665
  using assms
haftmann@49948
   666
proof -
haftmann@49948
   667
  have "\<exists>n. f n = f (Suc n)"
haftmann@49948
   668
  proof (rule ccontr)
haftmann@49948
   669
    assume "\<not> ?thesis"
haftmann@49948
   670
    then have "\<And>n. f n \<noteq> f (Suc n)" by auto
haftmann@49948
   671
    then have "\<And>n. f n < f (Suc n)"
haftmann@49948
   672
      using  `mono f` by (auto simp: le_less mono_iff_le_Suc)
haftmann@49948
   673
    with lift_Suc_mono_less_iff[of f]
haftmann@49948
   674
    have "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
haftmann@49948
   675
    then have "inj f"
haftmann@49948
   676
      by (auto simp: inj_on_def) (metis linorder_less_linear order_less_imp_not_eq)
haftmann@49948
   677
    with `finite (range f)` have "finite (UNIV::nat set)"
haftmann@49948
   678
      by (rule finite_imageD)
haftmann@49948
   679
    then show False by simp
haftmann@49948
   680
  qed
haftmann@49948
   681
  then obtain n where n: "f n = f (Suc n)" ..
haftmann@49948
   682
  def N \<equiv> "LEAST n. f n = f (Suc n)"
haftmann@49948
   683
  have N: "f N = f (Suc N)"
haftmann@49948
   684
    unfolding N_def using n by (rule LeastI)
haftmann@49948
   685
  show ?thesis
haftmann@49948
   686
  proof (intro exI[of _ N] conjI allI impI)
haftmann@49948
   687
    fix n assume "N \<le> n"
haftmann@49948
   688
    then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
haftmann@49948
   689
    proof (induct rule: dec_induct)
haftmann@49948
   690
      case (step n) then show ?case
haftmann@49948
   691
        using eq[rule_format, of "n - 1"] N
haftmann@49948
   692
        by (cases n) (auto simp add: le_Suc_eq)
haftmann@49948
   693
    qed simp
haftmann@49948
   694
    from this[of n] `N \<le> n` show "f N = f n" by auto
haftmann@49948
   695
  next
haftmann@49948
   696
    fix n m :: nat assume "m < n" "n \<le> N"
haftmann@49948
   697
    then show "f m < f n"
haftmann@49948
   698
    proof (induct rule: less_Suc_induct[consumes 1])
haftmann@49948
   699
      case (1 i)
haftmann@49948
   700
      then have "i < N" by simp
haftmann@49948
   701
      then have "f i \<noteq> f (Suc i)"
haftmann@49948
   702
        unfolding N_def by (rule not_less_Least)
haftmann@49948
   703
      with `mono f` show ?case by (simp add: mono_iff_le_Suc less_le)
haftmann@49948
   704
    qed auto
haftmann@49948
   705
  qed
haftmann@49948
   706
qed
haftmann@49948
   707
haftmann@49948
   708
lemma finite_mono_strict_prefix_implies_finite_fixpoint:
haftmann@49948
   709
  fixes f :: "nat \<Rightarrow> 'a set"
haftmann@49948
   710
  assumes S: "\<And>i. f i \<subseteq> S" "finite S"
haftmann@49948
   711
    and inj: "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n) \<and> (\<forall>n\<ge>N. f N = f n)"
haftmann@49948
   712
  shows "f (card S) = (\<Union>n. f n)"
haftmann@49948
   713
proof -
haftmann@49948
   714
  from inj obtain N where inj: "(\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n)" and eq: "(\<forall>n\<ge>N. f N = f n)" by auto
haftmann@49948
   715
haftmann@49948
   716
  { fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
haftmann@49948
   717
    proof (induct i)
haftmann@49948
   718
      case 0 then show ?case by simp
haftmann@49948
   719
    next
haftmann@49948
   720
      case (Suc i)
haftmann@49948
   721
      with inj[rule_format, of "Suc i" i]
haftmann@49948
   722
      have "(f i) \<subset> (f (Suc i))" by auto
haftmann@49948
   723
      moreover have "finite (f (Suc i))" using S by (rule finite_subset)
haftmann@49948
   724
      ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
haftmann@49948
   725
      with Suc show ?case using inj by auto
haftmann@49948
   726
    qed
haftmann@49948
   727
  }
haftmann@49948
   728
  then have "N \<le> card (f N)" by simp
haftmann@49948
   729
  also have "\<dots> \<le> card S" using S by (intro card_mono)
haftmann@49948
   730
  finally have "f (card S) = f N" using eq by auto
haftmann@49948
   731
  then show ?thesis using eq inj[rule_format, of N]
haftmann@49948
   732
    apply auto
haftmann@49948
   733
    apply (case_tac "n < N")
haftmann@49948
   734
    apply (auto simp: not_less)
haftmann@49948
   735
    done
haftmann@49948
   736
qed
haftmann@49948
   737
haftmann@49948
   738
primrec bacc :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a set" 
haftmann@49948
   739
where
haftmann@49948
   740
  "bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"
haftmann@49948
   741
| "bacc r (Suc n) = (bacc r n \<union> {x. \<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r n})"
haftmann@49948
   742
haftmann@49948
   743
lemma bacc_subseteq_acc:
haftmann@49948
   744
  "bacc r n \<subseteq> acc r"
haftmann@49948
   745
  by (induct n) (auto intro: acc.intros)
haftmann@49948
   746
haftmann@49948
   747
lemma bacc_mono:
haftmann@49948
   748
  "n \<le> m \<Longrightarrow> bacc r n \<subseteq> bacc r m"
haftmann@49948
   749
  by (induct rule: dec_induct) auto
haftmann@49948
   750
  
haftmann@49948
   751
lemma bacc_upper_bound:
haftmann@49948
   752
  "bacc (r :: ('a \<times> 'a) set)  (card (UNIV :: 'a::finite set)) = (\<Union>n. bacc r n)"
haftmann@49948
   753
proof -
haftmann@49948
   754
  have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
haftmann@49948
   755
  moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
haftmann@49948
   756
  moreover have "finite (range (bacc r))" by auto
haftmann@49948
   757
  ultimately show ?thesis
haftmann@49948
   758
   by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
haftmann@49948
   759
     (auto intro: finite_mono_remains_stable_implies_strict_prefix)
haftmann@49948
   760
qed
haftmann@49948
   761
haftmann@49948
   762
lemma acc_subseteq_bacc:
haftmann@49948
   763
  assumes "finite r"
haftmann@49948
   764
  shows "acc r \<subseteq> (\<Union>n. bacc r n)"
haftmann@49948
   765
proof
haftmann@49948
   766
  fix x
haftmann@49948
   767
  assume "x : acc r"
haftmann@49948
   768
  then have "\<exists> n. x : bacc r n"
haftmann@49948
   769
  proof (induct x arbitrary: rule: acc.induct)
haftmann@49948
   770
    case (accI x)
haftmann@49948
   771
    then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp
haftmann@49948
   772
    from choice[OF this] obtain n where n: "\<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r (n y)" ..
haftmann@49948
   773
    obtain n where "\<And>y. (y, x) : r \<Longrightarrow> y : bacc r n"
haftmann@49948
   774
    proof
haftmann@49948
   775
      fix y assume y: "(y, x) : r"
haftmann@49948
   776
      with n have "y : bacc r (n y)" by auto
haftmann@49948
   777
      moreover have "n y <= Max ((%(y, x). n y) ` r)"
haftmann@49948
   778
        using y `finite r` by (auto intro!: Max_ge)
haftmann@49948
   779
      note bacc_mono[OF this, of r]
haftmann@49948
   780
      ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto
haftmann@49948
   781
    qed
haftmann@49948
   782
    then show ?case
haftmann@49948
   783
      by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
haftmann@49948
   784
  qed
haftmann@49948
   785
  then show "x : (UN n. bacc r n)" by auto
haftmann@49948
   786
qed
haftmann@49948
   787
haftmann@49948
   788
lemma acc_bacc_eq:
haftmann@49948
   789
  fixes A :: "('a :: finite \<times> 'a) set"
haftmann@49948
   790
  assumes "finite A"
haftmann@49948
   791
  shows "acc A = bacc A (card (UNIV :: 'a set))"
haftmann@49948
   792
  using assms by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound order_eq_iff)
haftmann@49948
   793
haftmann@49948
   794
wenzelm@17893
   795
subsection {* Specification package -- Hilbertized version *}
wenzelm@17893
   796
wenzelm@17893
   797
lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
wenzelm@17893
   798
  by (simp only: someI_ex)
wenzelm@17893
   799
wenzelm@48891
   800
ML_file "Tools/choice_specification.ML"
skalberg@14115
   801
paulson@11451
   802
end
haftmann@49948
   803