src/HOL/Hilbert_Choice.thy
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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 Lattices_Big Metis
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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 _ => 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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text {*
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  Every infinite set contains a countable subset. More precisely we
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  show that a set @{text S} is infinite if and only if there exists an
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  injective function from the naturals into @{text S}.
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  The ``only if'' direction is harder because it requires the
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  construction of a sequence of pairwise different elements of an
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  infinite set @{text S}. The idea is to construct a sequence of
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  non-empty and infinite subsets of @{text S} obtained by successively
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  removing elements of @{text S}.
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*}
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lemma infinite_countable_subset:
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  assumes inf: "\<not> finite (S::'a set)"
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  shows "\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S"
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  -- {* Courtesy of Stephan Merz *}
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proof -
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   292
  def Sseq \<equiv> "nat_rec S (\<lambda>n T. T - {SOME e. e \<in> T})"
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   293
  def pick \<equiv> "\<lambda>n. (SOME e. e \<in> Sseq n)"
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   294
  { fix n have "Sseq n \<subseteq> S" "\<not> finite (Sseq n)" by (induct n) (auto simp add: Sseq_def inf) }
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   295
  moreover then have *: "\<And>n. pick n \<in> Sseq n" by (metis someI_ex pick_def ex_in_conv finite.simps)
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   296
  ultimately have "range pick \<subseteq> S" by auto
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   297
  moreover
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   298
  { fix n m                 
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   299
    have "pick n \<notin> Sseq (n + Suc m)" by (induct m) (auto simp add: Sseq_def pick_def)
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   300
    hence "pick n \<noteq> pick (n + Suc m)" by (metis *)
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   301
  }
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   302
  then have "inj pick" by (intro linorder_injI) (auto simp add: less_iff_Suc_add)
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   303
  ultimately show ?thesis by blast
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   304
qed
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   305
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   306
lemma infinite_iff_countable_subset: "\<not> finite S \<longleftrightarrow> (\<exists>f. inj (f::nat \<Rightarrow> 'a) \<and> range f \<subseteq> S)"
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   307
  -- {* Courtesy of Stephan Merz *}
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   308
  by (metis finite_imageD finite_subset infinite_UNIV_char_0 infinite_countable_subset)
9387251b6a46 eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents: 54295
diff changeset
   309
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   310
lemma image_inv_into_cancel:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   311
  assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   312
  shows "f `((inv_into A f)`B') = B'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   313
  using assms
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   314
proof (auto simp add: f_inv_into_f)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   315
  let ?f' = "(inv_into A f)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   316
  fix a' assume *: "a' \<in> B'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   317
  then have "a' \<in> A'" using SUB by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   318
  then have "a' = f (?f' a')"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   319
    using SURJ by (auto simp add: f_inv_into_f)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   320
  then show "a' \<in> f ` (?f' ` B')" using * by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   321
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   322
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   323
lemma inv_into_inv_into_eq:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   324
  assumes "bij_betw f A A'" "a \<in> A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   325
  shows "inv_into A' (inv_into A f) a = f a"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   326
proof -
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   327
  let ?f' = "inv_into A f"   let ?f'' = "inv_into A' ?f'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   328
  have 1: "bij_betw ?f' A' A" using assms
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   329
  by (auto simp add: bij_betw_inv_into)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   330
  obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   331
    using 1 `a \<in> A` unfolding bij_betw_def by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   332
  hence "?f'' a = a'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   333
    using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   334
  moreover have "f a = a'" using assms 2 3
44921
58eef4843641 tuned proofs
huffman
parents: 44890
diff changeset
   335
    by (auto simp add: bij_betw_def)
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   336
  ultimately show "?f'' a = f a" by simp
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   337
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   338
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   339
lemma inj_on_iff_surj:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   340
  assumes "A \<noteq> {}"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   341
  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   342
proof safe
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   343
  fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   344
  let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"  let ?csi = "\<lambda>a. a \<in> A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   345
  let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   346
  have "?g ` A' = A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   347
  proof
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   348
    show "?g ` A' \<le> A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   349
    proof clarify
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   350
      fix a' assume *: "a' \<in> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   351
      show "?g a' \<in> A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   352
      proof cases
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   353
        assume Case1: "a' \<in> f ` A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   354
        then obtain a where "?phi a' a" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   355
        hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   356
        with Case1 show ?thesis by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   357
      next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   358
        assume Case2: "a' \<notin> f ` A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   359
        hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   360
        with Case2 show ?thesis by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   361
      qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   362
    qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   363
  next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   364
    show "A \<le> ?g ` A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   365
    proof-
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   366
      {fix a assume *: "a \<in> A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   367
       let ?b = "SOME aa. ?phi (f a) aa"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   368
       have "?phi (f a) a" using * by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   369
       hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   370
       hence "?g(f a) = ?b" using * by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   371
       moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   372
       ultimately have "?g(f a) = a" by simp
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   373
       with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   374
      }
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   375
      thus ?thesis by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   376
    qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   377
  qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   378
  thus "\<exists>g. g ` A' = A" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   379
next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   380
  fix g  let ?f = "inv_into A' g"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   381
  have "inj_on ?f (g ` A')"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   382
    by (auto simp add: inj_on_inv_into)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   383
  moreover
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   384
  {fix a' assume *: "a' \<in> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   385
   let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   386
   have "?phi a'" using * by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   387
   hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   388
   hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   389
  }
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   390
  ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   391
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   392
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   393
lemma Ex_inj_on_UNION_Sigma:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   394
  "\<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))"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   395
proof
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   396
  let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   397
  let ?sm = "\<lambda> a. SOME i. ?phi a i"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   398
  let ?f = "\<lambda>a. (?sm a, a)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   399
  have "inj_on ?f (\<Union> i \<in> I. A i)" unfolding inj_on_def by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   400
  moreover
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   401
  { { fix i a assume "i \<in> I" and "a \<in> A i"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   402
      hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   403
    }
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   404
    hence "?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   405
  }
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   406
  ultimately
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   407
  show "inj_on ?f (\<Union> i \<in> I. A i) \<and> ?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   408
  by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   409
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   410
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   411
subsection {* The Cantor-Bernstein Theorem *}
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   412
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   413
lemma Cantor_Bernstein_aux:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   414
  shows "\<exists>A' h. A' \<le> A \<and>
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   415
                (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   416
                (\<forall>a \<in> A'. h a = f a) \<and>
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   417
                (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   418
proof-
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   419
  obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   420
  have 0: "mono H" unfolding mono_def H_def by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   421
  then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   422
  hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   423
  hence 3: "A' \<le> A" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   424
  have 4: "\<forall>a \<in> A'.  a \<notin> g`(B - f ` A')"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   425
  using 2 by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   426
  have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   427
  using 2 by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   428
  (*  *)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   429
  obtain h where h_def:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   430
  "h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   431
  hence "\<forall>a \<in> A'. h a = f a" by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   432
  moreover
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   433
  have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   434
  proof
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   435
    fix a assume *: "a \<in> A - A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   436
    let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   437
    have "h a = (SOME b. ?phi b)" using h_def * by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   438
    moreover have "\<exists>b. ?phi b" using 5 *  by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   439
    ultimately show  "?phi (h a)" using someI_ex[of ?phi] by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   440
  qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   441
  ultimately show ?thesis using 3 4 by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   442
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   443
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   444
theorem Cantor_Bernstein:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   445
  assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   446
          INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   447
  shows "\<exists>h. bij_betw h A B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   448
proof-
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   449
  obtain A' and h where 0: "A' \<le> A" and
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   450
  1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   451
  2: "\<forall>a \<in> A'. h a = f a" and
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   452
  3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   453
  using Cantor_Bernstein_aux[of A g B f] by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   454
  have "inj_on h A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   455
  proof (intro inj_onI)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   456
    fix a1 a2
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   457
    assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   458
    show "a1 = a2"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   459
    proof(cases "a1 \<in> A'")
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   460
      assume Case1: "a1 \<in> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   461
      show ?thesis
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   462
      proof(cases "a2 \<in> A'")
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   463
        assume Case11: "a2 \<in> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   464
        hence "f a1 = f a2" using Case1 2 6 by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   465
        thus ?thesis using INJ1 Case1 Case11 0
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   466
        unfolding inj_on_def by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   467
      next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   468
        assume Case12: "a2 \<notin> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   469
        hence False using 3 5 2 6 Case1 by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   470
        thus ?thesis by simp
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   471
      qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   472
    next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   473
    assume Case2: "a1 \<notin> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   474
      show ?thesis
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   475
      proof(cases "a2 \<in> A'")
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   476
        assume Case21: "a2 \<in> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   477
        hence False using 3 4 2 6 Case2 by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   478
        thus ?thesis by simp
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   479
      next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   480
        assume Case22: "a2 \<notin> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   481
        hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   482
        thus ?thesis using 6 by simp
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   483
      qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   484
    qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   485
  qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   486
  (*  *)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   487
  moreover
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   488
  have "h ` A = B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   489
  proof safe
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   490
    fix a assume "a \<in> A"
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 46950
diff changeset
   491
    thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   492
  next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   493
    fix b assume *: "b \<in> B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   494
    show "b \<in> h ` A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   495
    proof(cases "b \<in> f ` A'")
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   496
      assume Case1: "b \<in> f ` A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   497
      then obtain a where "a \<in> A' \<and> b = f a" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   498
      thus ?thesis using 2 0 by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   499
    next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   500
      assume Case2: "b \<notin> f ` A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   501
      hence "g b \<notin> A'" using 1 * by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   502
      hence 4: "g b \<in> A - A'" using * SUB2 by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   503
      hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   504
      using 3 by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   505
      hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   506
      thus ?thesis using 4 by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   507
    qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   508
  qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   509
  (*  *)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   510
  ultimately show ?thesis unfolding bij_betw_def by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   511
qed
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   512
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   513
subsection {*Other Consequences of Hilbert's Epsilon*}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   514
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   515
text {*Hilbert's Epsilon and the @{term split} Operator*}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   516
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   517
text{*Looping simprule*}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   518
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
26347
105f55201077 tuned proofs
haftmann
parents: 26105
diff changeset
   519
  by simp
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   520
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   521
lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
26347
105f55201077 tuned proofs
haftmann
parents: 26105
diff changeset
   522
  by (simp add: split_def)
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   523
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   524
lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
26347
105f55201077 tuned proofs
haftmann
parents: 26105
diff changeset
   525
  by blast
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   526
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   527
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   528
text{*A relation is wellfounded iff it has no infinite descending chain*}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   529
lemma wf_iff_no_infinite_down_chain:
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   530
  "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   531
apply (simp only: wf_eq_minimal)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   532
apply (rule iffI)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   533
 apply (rule notI)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   534
 apply (erule exE)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   535
 apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   536
apply (erule contrapos_np, simp, clarify)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   537
apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   538
 apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   539
 apply (rule allI, simp)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   540
 apply (rule someI2_ex, blast, blast)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   541
apply (rule allI)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   542
apply (induct_tac "n", simp_all)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   543
apply (rule someI2_ex, blast+)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   544
done
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   545
27760
3aa86edac080 added lemma
nipkow
parents: 26748
diff changeset
   546
lemma wf_no_infinite_down_chainE:
3aa86edac080 added lemma
nipkow
parents: 26748
diff changeset
   547
  assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
3aa86edac080 added lemma
nipkow
parents: 26748
diff changeset
   548
using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
3aa86edac080 added lemma
nipkow
parents: 26748
diff changeset
   549
3aa86edac080 added lemma
nipkow
parents: 26748
diff changeset
   550
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   551
text{*A dynamically-scoped fact for TFL *}
12298
wenzelm
parents: 12023
diff changeset
   552
lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
wenzelm
parents: 12023
diff changeset
   553
  by (blast intro: someI)
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   554
12298
wenzelm
parents: 12023
diff changeset
   555
wenzelm
parents: 12023
diff changeset
   556
subsection {* Least value operator *}
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   557
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35216
diff changeset
   558
definition
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35216
diff changeset
   559
  LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   560
  "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   561
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   562
syntax
12298
wenzelm
parents: 12023
diff changeset
   563
  "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   564
translations
35115
446c5063e4fd modernized translations;
wenzelm
parents: 33057
diff changeset
   565
  "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   566
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   567
lemma LeastMI2:
12298
wenzelm
parents: 12023
diff changeset
   568
  "P x ==> (!!y. P y ==> m x <= m y)
wenzelm
parents: 12023
diff changeset
   569
    ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
wenzelm
parents: 12023
diff changeset
   570
    ==> Q (LeastM m P)"
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   571
  apply (simp add: LeastM_def)
14208
144f45277d5a misc tidying
paulson
parents: 14115
diff changeset
   572
  apply (rule someI2_ex, blast, blast)
12298
wenzelm
parents: 12023
diff changeset
   573
  done
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   574
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   575
lemma LeastM_equality:
12298
wenzelm
parents: 12023
diff changeset
   576
  "P k ==> (!!x. P x ==> m k <= m x)
wenzelm
parents: 12023
diff changeset
   577
    ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
14208
144f45277d5a misc tidying
paulson
parents: 14115
diff changeset
   578
  apply (rule LeastMI2, assumption, blast)
12298
wenzelm
parents: 12023
diff changeset
   579
  apply (blast intro!: order_antisym)
wenzelm
parents: 12023
diff changeset
   580
  done
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   581
11454
7514e5e21cb8 Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents: 11451
diff changeset
   582
lemma wf_linord_ex_has_least:
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   583
  "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   584
    ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
12298
wenzelm
parents: 12023
diff changeset
   585
  apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
14208
144f45277d5a misc tidying
paulson
parents: 14115
diff changeset
   586
  apply (drule_tac x = "m`Collect P" in spec, force)
12298
wenzelm
parents: 12023
diff changeset
   587
  done
11454
7514e5e21cb8 Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents: 11451
diff changeset
   588
7514e5e21cb8 Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents: 11451
diff changeset
   589
lemma ex_has_least_nat:
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   590
    "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
12298
wenzelm
parents: 12023
diff changeset
   591
  apply (simp only: pred_nat_trancl_eq_le [symmetric])
wenzelm
parents: 12023
diff changeset
   592
  apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16563
diff changeset
   593
   apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
12298
wenzelm
parents: 12023
diff changeset
   594
  done
11454
7514e5e21cb8 Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents: 11451
diff changeset
   595
12298
wenzelm
parents: 12023
diff changeset
   596
lemma LeastM_nat_lemma:
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   597
    "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   598
  apply (simp add: LeastM_def)
12298
wenzelm
parents: 12023
diff changeset
   599
  apply (rule someI_ex)
wenzelm
parents: 12023
diff changeset
   600
  apply (erule ex_has_least_nat)
wenzelm
parents: 12023
diff changeset
   601
  done
11454
7514e5e21cb8 Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents: 11451
diff changeset
   602
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 44921
diff changeset
   603
lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1]
11454
7514e5e21cb8 Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents: 11451
diff changeset
   604
7514e5e21cb8 Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents: 11451
diff changeset
   605
lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
14208
144f45277d5a misc tidying
paulson
parents: 14115
diff changeset
   606
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
11454
7514e5e21cb8 Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
paulson
parents: 11451
diff changeset
   607
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   608
12298
wenzelm
parents: 12023
diff changeset
   609
subsection {* Greatest value operator *}
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   610
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35216
diff changeset
   611
definition
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35216
diff changeset
   612
  GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   613
  "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
12298
wenzelm
parents: 12023
diff changeset
   614
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35216
diff changeset
   615
definition
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35216
diff changeset
   616
  Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
12298
wenzelm
parents: 12023
diff changeset
   617
  "Greatest == GreatestM (%x. x)"
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   618
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   619
syntax
35115
446c5063e4fd modernized translations;
wenzelm
parents: 33057
diff changeset
   620
  "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
12298
wenzelm
parents: 12023
diff changeset
   621
      ("GREATEST _ WRT _. _" [0, 4, 10] 10)
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   622
translations
35115
446c5063e4fd modernized translations;
wenzelm
parents: 33057
diff changeset
   623
  "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   624
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   625
lemma GreatestMI2:
12298
wenzelm
parents: 12023
diff changeset
   626
  "P x ==> (!!y. P y ==> m y <= m x)
wenzelm
parents: 12023
diff changeset
   627
    ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
wenzelm
parents: 12023
diff changeset
   628
    ==> Q (GreatestM m P)"
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   629
  apply (simp add: GreatestM_def)
14208
144f45277d5a misc tidying
paulson
parents: 14115
diff changeset
   630
  apply (rule someI2_ex, blast, blast)
12298
wenzelm
parents: 12023
diff changeset
   631
  done
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   632
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   633
lemma GreatestM_equality:
12298
wenzelm
parents: 12023
diff changeset
   634
 "P k ==> (!!x. P x ==> m x <= m k)
wenzelm
parents: 12023
diff changeset
   635
    ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
14208
144f45277d5a misc tidying
paulson
parents: 14115
diff changeset
   636
  apply (rule_tac m = m in GreatestMI2, assumption, blast)
12298
wenzelm
parents: 12023
diff changeset
   637
  apply (blast intro!: order_antisym)
wenzelm
parents: 12023
diff changeset
   638
  done
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   639
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   640
lemma Greatest_equality:
12298
wenzelm
parents: 12023
diff changeset
   641
  "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   642
  apply (simp add: Greatest_def)
14208
144f45277d5a misc tidying
paulson
parents: 14115
diff changeset
   643
  apply (erule GreatestM_equality, blast)
12298
wenzelm
parents: 12023
diff changeset
   644
  done
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   645
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   646
lemma ex_has_greatest_nat_lemma:
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   647
  "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   648
    ==> \<exists>y. P y & ~ (m y < m k + n)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   649
  apply (induct n, force)
12298
wenzelm
parents: 12023
diff changeset
   650
  apply (force simp add: le_Suc_eq)
wenzelm
parents: 12023
diff changeset
   651
  done
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   652
12298
wenzelm
parents: 12023
diff changeset
   653
lemma ex_has_greatest_nat:
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   654
  "P k ==> \<forall>y. P y --> m y < b
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   655
    ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
12298
wenzelm
parents: 12023
diff changeset
   656
  apply (rule ccontr)
wenzelm
parents: 12023
diff changeset
   657
  apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
14208
144f45277d5a misc tidying
paulson
parents: 14115
diff changeset
   658
    apply (subgoal_tac [3] "m k <= b", auto)
12298
wenzelm
parents: 12023
diff changeset
   659
  done
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   660
12298
wenzelm
parents: 12023
diff changeset
   661
lemma GreatestM_nat_lemma:
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   662
  "P k ==> \<forall>y. P y --> m y < b
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   663
    ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   664
  apply (simp add: GreatestM_def)
12298
wenzelm
parents: 12023
diff changeset
   665
  apply (rule someI_ex)
14208
144f45277d5a misc tidying
paulson
parents: 14115
diff changeset
   666
  apply (erule ex_has_greatest_nat, assumption)
12298
wenzelm
parents: 12023
diff changeset
   667
  done
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   668
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 44921
diff changeset
   669
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   670
12298
wenzelm
parents: 12023
diff changeset
   671
lemma GreatestM_nat_le:
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   672
  "P x ==> \<forall>y. P y --> m y < b
12298
wenzelm
parents: 12023
diff changeset
   673
    ==> (m x::nat) <= m (GreatestM m P)"
21020
9af9ceb16d58 Adapted to changes in FixedPoint theory.
berghofe
parents: 18389
diff changeset
   674
  apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
12298
wenzelm
parents: 12023
diff changeset
   675
  done
wenzelm
parents: 12023
diff changeset
   676
wenzelm
parents: 12023
diff changeset
   677
wenzelm
parents: 12023
diff changeset
   678
text {* \medskip Specialization to @{text GREATEST}. *}
wenzelm
parents: 12023
diff changeset
   679
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   680
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   681
  apply (simp add: Greatest_def)
14208
144f45277d5a misc tidying
paulson
parents: 14115
diff changeset
   682
  apply (rule GreatestM_natI, auto)
12298
wenzelm
parents: 12023
diff changeset
   683
  done
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   684
12298
wenzelm
parents: 12023
diff changeset
   685
lemma Greatest_le:
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   686
    "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   687
  apply (simp add: Greatest_def)
14208
144f45277d5a misc tidying
paulson
parents: 14115
diff changeset
   688
  apply (rule GreatestM_nat_le, auto)
12298
wenzelm
parents: 12023
diff changeset
   689
  done
wenzelm
parents: 12023
diff changeset
   690
wenzelm
parents: 12023
diff changeset
   691
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   692
subsection {* An aside: bounded accessible part *}
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   693
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   694
text {* Finite monotone eventually stable sequences *}
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   695
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   696
lemma finite_mono_remains_stable_implies_strict_prefix:
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   697
  fixes f :: "nat \<Rightarrow> 'a::order"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   698
  assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   699
  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)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   700
  using assms
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   701
proof -
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   702
  have "\<exists>n. f n = f (Suc n)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   703
  proof (rule ccontr)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   704
    assume "\<not> ?thesis"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   705
    then have "\<And>n. f n \<noteq> f (Suc n)" by auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   706
    then have "\<And>n. f n < f (Suc n)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   707
      using  `mono f` by (auto simp: le_less mono_iff_le_Suc)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   708
    with lift_Suc_mono_less_iff[of f]
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   709
    have "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   710
    then have "inj f"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   711
      by (auto simp: inj_on_def) (metis linorder_less_linear order_less_imp_not_eq)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   712
    with `finite (range f)` have "finite (UNIV::nat set)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   713
      by (rule finite_imageD)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   714
    then show False by simp
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   715
  qed
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   716
  then obtain n where n: "f n = f (Suc n)" ..
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   717
  def N \<equiv> "LEAST n. f n = f (Suc n)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   718
  have N: "f N = f (Suc N)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   719
    unfolding N_def using n by (rule LeastI)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   720
  show ?thesis
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   721
  proof (intro exI[of _ N] conjI allI impI)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   722
    fix n assume "N \<le> n"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   723
    then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   724
    proof (induct rule: dec_induct)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   725
      case (step n) then show ?case
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   726
        using eq[rule_format, of "n - 1"] N
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   727
        by (cases n) (auto simp add: le_Suc_eq)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   728
    qed simp
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   729
    from this[of n] `N \<le> n` show "f N = f n" by auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   730
  next
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   731
    fix n m :: nat assume "m < n" "n \<le> N"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   732
    then show "f m < f n"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   733
    proof (induct rule: less_Suc_induct[consumes 1])
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   734
      case (1 i)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   735
      then have "i < N" by simp
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   736
      then have "f i \<noteq> f (Suc i)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   737
        unfolding N_def by (rule not_less_Least)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   738
      with `mono f` show ?case by (simp add: mono_iff_le_Suc less_le)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   739
    qed auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   740
  qed
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   741
qed
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   742
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   743
lemma finite_mono_strict_prefix_implies_finite_fixpoint:
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   744
  fixes f :: "nat \<Rightarrow> 'a set"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   745
  assumes S: "\<And>i. f i \<subseteq> S" "finite S"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   746
    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)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   747
  shows "f (card S) = (\<Union>n. f n)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   748
proof -
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   749
  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
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   750
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   751
  { fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   752
    proof (induct i)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   753
      case 0 then show ?case by simp
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   754
    next
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   755
      case (Suc i)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   756
      with inj[rule_format, of "Suc i" i]
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   757
      have "(f i) \<subset> (f (Suc i))" by auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   758
      moreover have "finite (f (Suc i))" using S by (rule finite_subset)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   759
      ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   760
      with Suc show ?case using inj by auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   761
    qed
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   762
  }
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   763
  then have "N \<le> card (f N)" by simp
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   764
  also have "\<dots> \<le> card S" using S by (intro card_mono)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   765
  finally have "f (card S) = f N" using eq by auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   766
  then show ?thesis using eq inj[rule_format, of N]
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   767
    apply auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   768
    apply (case_tac "n < N")
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   769
    apply (auto simp: not_less)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   770
    done
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   771
qed
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   772
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   773
primrec bacc :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a set" 
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   774
where
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   775
  "bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   776
| "bacc r (Suc n) = (bacc r n \<union> {x. \<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r n})"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   777
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   778
lemma bacc_subseteq_acc:
54295
45a5523d4a63 qualifed popular user space names
haftmann
parents: 52143
diff changeset
   779
  "bacc r n \<subseteq> Wellfounded.acc r"
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   780
  by (induct n) (auto intro: acc.intros)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   781
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   782
lemma bacc_mono:
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   783
  "n \<le> m \<Longrightarrow> bacc r n \<subseteq> bacc r m"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   784
  by (induct rule: dec_induct) auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   785
  
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   786
lemma bacc_upper_bound:
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   787
  "bacc (r :: ('a \<times> 'a) set)  (card (UNIV :: 'a::finite set)) = (\<Union>n. bacc r n)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   788
proof -
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   789
  have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   790
  moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   791
  moreover have "finite (range (bacc r))" by auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   792
  ultimately show ?thesis
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   793
   by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   794
     (auto intro: finite_mono_remains_stable_implies_strict_prefix)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   795
qed
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   796
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   797
lemma acc_subseteq_bacc:
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   798
  assumes "finite r"
54295
45a5523d4a63 qualifed popular user space names
haftmann
parents: 52143
diff changeset
   799
  shows "Wellfounded.acc r \<subseteq> (\<Union>n. bacc r n)"
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   800
proof
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   801
  fix x
54295
45a5523d4a63 qualifed popular user space names
haftmann
parents: 52143
diff changeset
   802
  assume "x : Wellfounded.acc r"
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   803
  then have "\<exists> n. x : bacc r n"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   804
  proof (induct x arbitrary: rule: acc.induct)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   805
    case (accI x)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   806
    then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   807
    from choice[OF this] obtain n where n: "\<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r (n y)" ..
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   808
    obtain n where "\<And>y. (y, x) : r \<Longrightarrow> y : bacc r n"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   809
    proof
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   810
      fix y assume y: "(y, x) : r"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   811
      with n have "y : bacc r (n y)" by auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   812
      moreover have "n y <= Max ((%(y, x). n y) ` r)"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   813
        using y `finite r` by (auto intro!: Max_ge)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   814
      note bacc_mono[OF this, of r]
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   815
      ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   816
    qed
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   817
    then show ?case
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   818
      by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   819
  qed
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   820
  then show "x : (UN n. bacc r n)" by auto
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   821
qed
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   822
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   823
lemma acc_bacc_eq:
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   824
  fixes A :: "('a :: finite \<times> 'a) set"
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   825
  assumes "finite A"
54295
45a5523d4a63 qualifed popular user space names
haftmann
parents: 52143
diff changeset
   826
  shows "Wellfounded.acc A = bacc A (card (UNIV :: 'a set))"
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   827
  using assms by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound order_eq_iff)
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   828
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   829
55020
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   830
subsection {* More on injections, bijections, and inverses *}
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   831
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   832
lemma infinite_imp_bij_betw:
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   833
assumes INF: "\<not> finite A"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   834
shows "\<exists>h. bij_betw h A (A - {a})"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   835
proof(cases "a \<in> A")
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   836
  assume Case1: "a \<notin> A"  hence "A - {a} = A" by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   837
  thus ?thesis using bij_betw_id[of A] by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   838
next
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   839
  assume Case2: "a \<in> A"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   840
find_theorems "\<not> finite _"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   841
  have "\<not> finite (A - {a})" using INF by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   842
  with infinite_iff_countable_subset[of "A - {a}"] obtain f::"nat \<Rightarrow> 'a"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   843
  where 1: "inj f" and 2: "f ` UNIV \<le> A - {a}" by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   844
  obtain g where g_def: "g = (\<lambda> n. if n = 0 then a else f (Suc n))" by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   845
  obtain A' where A'_def: "A' = g ` UNIV" by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   846
  have temp: "\<forall>y. f y \<noteq> a" using 2 by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   847
  have 3: "inj_on g UNIV \<and> g ` UNIV \<le> A \<and> a \<in> g ` UNIV"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   848
  proof(auto simp add: Case2 g_def, unfold inj_on_def, intro ballI impI,
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   849
        case_tac "x = 0", auto simp add: 2)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   850
    fix y  assume "a = (if y = 0 then a else f (Suc y))"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   851
    thus "y = 0" using temp by (case_tac "y = 0", auto)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   852
  next
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   853
    fix x y
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   854
    assume "f (Suc x) = (if y = 0 then a else f (Suc y))"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   855
    thus "x = y" using 1 temp unfolding inj_on_def by (case_tac "y = 0", auto)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   856
  next
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   857
    fix n show "f (Suc n) \<in> A" using 2 by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   858
  qed
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   859
  hence 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<le> A"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   860
  using inj_on_imp_bij_betw[of g] unfolding A'_def by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   861
  hence 5: "bij_betw (inv g) A' UNIV"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   862
  by (auto simp add: bij_betw_inv_into)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   863
  (*  *)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   864
  obtain n where "g n = a" using 3 by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   865
  hence 6: "bij_betw g (UNIV - {n}) (A' - {a})"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   866
  using 3 4 unfolding A'_def
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   867
  by clarify (rule bij_betw_subset, auto simp: image_set_diff)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   868
  (*  *)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   869
  obtain v where v_def: "v = (\<lambda> m. if m < n then m else Suc m)" by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   870
  have 7: "bij_betw v UNIV (UNIV - {n})"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   871
  proof(unfold bij_betw_def inj_on_def, intro conjI, clarify)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   872
    fix m1 m2 assume "v m1 = v m2"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   873
    thus "m1 = m2"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   874
    by(case_tac "m1 < n", case_tac "m2 < n",
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   875
       auto simp add: inj_on_def v_def, case_tac "m2 < n", auto)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   876
  next
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   877
    show "v ` UNIV = UNIV - {n}"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   878
    proof(auto simp add: v_def)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   879
      fix m assume *: "m \<noteq> n" and **: "m \<notin> Suc ` {m'. \<not> m' < n}"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   880
      {assume "n \<le> m" with * have 71: "Suc n \<le> m" by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   881
       then obtain m' where 72: "m = Suc m'" using Suc_le_D by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   882
       with 71 have "n \<le> m'" by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   883
       with 72 ** have False by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   884
      }
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   885
      thus "m < n" by force
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   886
    qed
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   887
  qed
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   888
  (*  *)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   889
  obtain h' where h'_def: "h' = g o v o (inv g)" by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   890
  hence 8: "bij_betw h' A' (A' - {a})" using 5 7 6
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   891
  by (auto simp add: bij_betw_trans)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   892
  (*  *)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   893
  obtain h where h_def: "h = (\<lambda> b. if b \<in> A' then h' b else b)" by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   894
  have "\<forall>b \<in> A'. h b = h' b" unfolding h_def by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   895
  hence "bij_betw h  A' (A' - {a})" using 8 bij_betw_cong[of A' h] by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   896
  moreover
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   897
  {have "\<forall>b \<in> A - A'. h b = b" unfolding h_def by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   898
   hence "bij_betw h  (A - A') (A - A')"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   899
   using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   900
  }
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   901
  moreover
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   902
  have "(A' Int (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   903
        ((A' - {a}) Int (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   904
  using 4 by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   905
  ultimately have "bij_betw h A (A - {a})"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   906
  using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   907
  thus ?thesis by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   908
qed
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   909
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   910
lemma infinite_imp_bij_betw2:
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   911
assumes INF: "\<not> finite A"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   912
shows "\<exists>h. bij_betw h A (A \<union> {a})"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   913
proof(cases "a \<in> A")
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   914
  assume Case1: "a \<in> A"  hence "A \<union> {a} = A" by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   915
  thus ?thesis using bij_betw_id[of A] by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   916
next
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   917
  let ?A' = "A \<union> {a}"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   918
  assume Case2: "a \<notin> A" hence "A = ?A' - {a}" by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   919
  moreover have "\<not> finite ?A'" using INF by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   920
  ultimately obtain f where "bij_betw f ?A' A"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   921
  using infinite_imp_bij_betw[of ?A' a] by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   922
  hence "bij_betw(inv_into ?A' f) A ?A'" using bij_betw_inv_into by blast
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   923
  thus ?thesis by auto
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   924
qed
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   925
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   926
lemma bij_betw_inv_into_left:
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   927
assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   928
shows "(inv_into A f) (f a) = a"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   929
using assms unfolding bij_betw_def
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   930
by clarify (rule inv_into_f_f)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   931
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   932
lemma bij_betw_inv_into_right:
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   933
assumes "bij_betw f A A'" "a' \<in> A'"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   934
shows "f(inv_into A f a') = a'"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   935
using assms unfolding bij_betw_def using f_inv_into_f by force
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   936
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   937
lemma bij_betw_inv_into_subset:
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   938
assumes BIJ: "bij_betw f A A'" and
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   939
        SUB: "B \<le> A" and IM: "f ` B = B'"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   940
shows "bij_betw (inv_into A f) B' B"
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   941
using assms unfolding bij_betw_def
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   942
by (auto intro: inj_on_inv_into)
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   943
96b05fd2aee4 dissolved 'Fun_More_FP' (a BNF dependency)
blanchet
parents: 54744
diff changeset
   944
17893
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   945
subsection {* Specification package -- Hilbertized version *}
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   946
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   947
lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   948
  by (simp only: someI_ex)
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   949
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47988
diff changeset
   950
ML_file "Tools/choice_specification.ML"
14115
65ec3f73d00b Added package for definition by specification.
skalberg
parents: 13764
diff changeset
   951
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   952
end
49948
744934b818c7 moved quite generic material from theory Enum to more appropriate places
haftmann
parents: 49739
diff changeset
   953