src/HOL/Hilbert_Choice.thy
author wenzelm
Fri Apr 08 13:31:16 2011 +0200 (2011-04-08)
changeset 42284 326f57825e1a
parent 40703 d1fc454d6735
child 44890 22f665a2e91c
permissions -rw-r--r--
explicit structure Syntax_Trans;
discontinued old-style constrainAbsC;
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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 Plain
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uses ("Tools/choice_specification.ML")
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begin
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subsection {* Hilbert's epsilon *}
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axiomatization Eps :: "('a => bool) => 'a" where
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  someI: "P x ==> P (Eps P)"
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syntax (epsilon)
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  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
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syntax (HOL)
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  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
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syntax
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  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
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translations
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  "SOME x. P" == "CONST Eps (%x. P)"
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print_translation {*
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  [(@{const_syntax Eps}, fn [Abs abs] =>
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      let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
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      in Syntax.const @{syntax_const "_Eps"} $ x $ t end)]
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*} -- {* to avoid eta-contraction of body *}
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definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
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"inv_into A f == %x. SOME y. y : A & f y = x"
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abbreviation inv :: "('a => 'b) => ('b => 'a)" where
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"inv == inv_into UNIV"
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subsection {*Hilbert's Epsilon-operator*}
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text{*Easier to apply than @{text someI} if the witness comes from an
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existential formula*}
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lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
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apply (erule exE)
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apply (erule someI)
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done
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text{*Easier to apply than @{text someI} because the conclusion has only one
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occurrence of @{term P}.*}
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lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
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by (blast intro: someI)
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text{*Easier to apply than @{text someI2} if the witness comes from an
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existential formula*}
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lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
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by (blast intro: someI2)
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lemma some_equality [intro]:
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     "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
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by (blast intro: someI2)
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lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
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by blast
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lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
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by (blast intro: someI)
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lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
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apply (rule some_equality)
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apply (rule refl, assumption)
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done
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lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
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apply (rule some_equality)
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apply (rule refl)
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apply (erule sym)
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done
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subsection{*Axiom of Choice, Proved Using the Description Operator*}
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lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
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by (fast elim: someI)
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lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
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by (fast elim: someI)
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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: ext 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: o_assoc[symmetric])
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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(fastsimp simp: image_def)
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lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
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by (blast intro!: surjI inv_into_f_f)
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lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
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by (simp add: f_inv_into_f)
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lemma inv_into_injective:
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  assumes eq: "inv_into A f x = inv_into A f y"
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      and x: "x: f`A"
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      and y: "y: f`A"
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  shows "x=y"
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proof -
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  have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
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  thus ?thesis by (simp add: f_inv_into_f x y)
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qed
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lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
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by (blast intro: inj_onI dest: inv_into_injective injD)
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lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
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by (auto simp add: bij_betw_def inj_on_inv_into)
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lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
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by (simp add: inj_on_inv_into)
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lemma surj_iff: "(surj f) = (f o inv f = id)"
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by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
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lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
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  unfolding surj_iff by (simp add: o_def fun_eq_iff)
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lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
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apply (rule ext)
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apply (drule_tac x = "inv f x" in spec)
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apply (simp add: surj_f_inv_f)
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done
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lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
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by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
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lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
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apply (rule ext)
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apply (auto simp add: inv_into_def)
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done
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lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
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apply (rule inv_equality)
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apply (auto simp add: bij_def surj_f_inv_f)
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done
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(** bij(inv f) implies little about f.  Consider f::bool=>bool such that
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    f(True)=f(False)=True.  Then it's consistent with axiom someI that
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    inv f could be any function at all, including the identity function.
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    If inv f=id then inv f is a bijection, but inj f, surj(f) and
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    inv(inv f)=f all fail.
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**)
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lemma inv_into_comp:
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  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
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  inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
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apply (rule inv_into_f_eq)
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  apply (fast intro: comp_inj_on)
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 apply (simp add: inv_into_into)
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apply (simp add: f_inv_into_f inv_into_into)
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done
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lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
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apply (rule inv_equality)
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apply (auto simp add: bij_def surj_f_inv_f)
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done
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lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
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by (simp add: image_eq_UN surj_f_inv_f)
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lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
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by (simp add: image_eq_UN)
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lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
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by (auto simp add: image_def)
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lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
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apply auto
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apply (force simp add: bij_is_inj)
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apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
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done
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lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A" 
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apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
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apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
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done
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lemma finite_fun_UNIVD1:
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  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
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  and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
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  shows "finite (UNIV :: 'a set)"
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proof -
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  from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
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  with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
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    by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
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  then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
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  then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
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  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
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  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
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  proof (rule UNIV_eq_I)
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    fix x :: 'a
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    from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
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    thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
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  qed
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  ultimately show "finite (UNIV :: 'a set)" by simp
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qed
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lemma image_inv_into_cancel:
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  assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
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  shows "f `((inv_into A f)`B') = B'"
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  using assms
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proof (auto simp add: f_inv_into_f)
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  let ?f' = "(inv_into A f)"
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  fix a' assume *: "a' \<in> B'"
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  then have "a' \<in> A'" using SUB by auto
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  then have "a' = f (?f' a')"
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    using SURJ by (auto simp add: f_inv_into_f)
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  then show "a' \<in> f ` (?f' ` B')" using * by blast
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qed
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lemma inv_into_inv_into_eq:
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  assumes "bij_betw f A A'" "a \<in> A"
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  shows "inv_into A' (inv_into A f) a = f a"
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proof -
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  let ?f' = "inv_into A f"   let ?f'' = "inv_into A' ?f'"
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  have 1: "bij_betw ?f' A' A" using assms
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  by (auto simp add: bij_betw_inv_into)
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  obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
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    using 1 `a \<in> A` unfolding bij_betw_def by force
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  hence "?f'' a = a'"
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    using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
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  moreover have "f a = a'" using assms 2 3
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    by (auto simp add: inv_into_f_f bij_betw_def)
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  ultimately show "?f'' a = f a" by simp
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qed
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lemma inj_on_iff_surj:
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  assumes "A \<noteq> {}"
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  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
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proof safe
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  fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
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  let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"  let ?csi = "\<lambda>a. a \<in> A"
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  let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
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  have "?g ` A' = A"
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  proof
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    show "?g ` A' \<le> A"
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    proof clarify
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      fix a' assume *: "a' \<in> A'"
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      show "?g a' \<in> A"
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      proof cases
hoelzl@40703
   307
        assume Case1: "a' \<in> f ` A"
hoelzl@40703
   308
        then obtain a where "?phi a' a" by blast
hoelzl@40703
   309
        hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
hoelzl@40703
   310
        with Case1 show ?thesis by auto
hoelzl@40703
   311
      next
hoelzl@40703
   312
        assume Case2: "a' \<notin> f ` A"
hoelzl@40703
   313
        hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
hoelzl@40703
   314
        with Case2 show ?thesis by auto
hoelzl@40703
   315
      qed
hoelzl@40703
   316
    qed
hoelzl@40703
   317
  next
hoelzl@40703
   318
    show "A \<le> ?g ` A'"
hoelzl@40703
   319
    proof-
hoelzl@40703
   320
      {fix a assume *: "a \<in> A"
hoelzl@40703
   321
       let ?b = "SOME aa. ?phi (f a) aa"
hoelzl@40703
   322
       have "?phi (f a) a" using * by auto
hoelzl@40703
   323
       hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
hoelzl@40703
   324
       hence "?g(f a) = ?b" using * by auto
hoelzl@40703
   325
       moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
hoelzl@40703
   326
       ultimately have "?g(f a) = a" by simp
hoelzl@40703
   327
       with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
hoelzl@40703
   328
      }
hoelzl@40703
   329
      thus ?thesis by force
hoelzl@40703
   330
    qed
hoelzl@40703
   331
  qed
hoelzl@40703
   332
  thus "\<exists>g. g ` A' = A" by blast
hoelzl@40703
   333
next
hoelzl@40703
   334
  fix g  let ?f = "inv_into A' g"
hoelzl@40703
   335
  have "inj_on ?f (g ` A')"
hoelzl@40703
   336
    by (auto simp add: inj_on_inv_into)
hoelzl@40703
   337
  moreover
hoelzl@40703
   338
  {fix a' assume *: "a' \<in> A'"
hoelzl@40703
   339
   let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
hoelzl@40703
   340
   have "?phi a'" using * by auto
hoelzl@40703
   341
   hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
hoelzl@40703
   342
   hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
hoelzl@40703
   343
  }
hoelzl@40703
   344
  ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
hoelzl@40703
   345
qed
hoelzl@40703
   346
hoelzl@40703
   347
lemma Ex_inj_on_UNION_Sigma:
hoelzl@40703
   348
  "\<exists>f. (inj_on f (\<Union> i \<in> I. A i) \<and> f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i))"
hoelzl@40703
   349
proof
hoelzl@40703
   350
  let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
hoelzl@40703
   351
  let ?sm = "\<lambda> a. SOME i. ?phi a i"
hoelzl@40703
   352
  let ?f = "\<lambda>a. (?sm a, a)"
hoelzl@40703
   353
  have "inj_on ?f (\<Union> i \<in> I. A i)" unfolding inj_on_def by auto
hoelzl@40703
   354
  moreover
hoelzl@40703
   355
  { { fix i a assume "i \<in> I" and "a \<in> A i"
hoelzl@40703
   356
      hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
hoelzl@40703
   357
    }
hoelzl@40703
   358
    hence "?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
hoelzl@40703
   359
  }
hoelzl@40703
   360
  ultimately
hoelzl@40703
   361
  show "inj_on ?f (\<Union> i \<in> I. A i) \<and> ?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)"
hoelzl@40703
   362
  by auto
hoelzl@40703
   363
qed
hoelzl@40703
   364
hoelzl@40703
   365
subsection {* The Cantor-Bernstein Theorem *}
hoelzl@40703
   366
hoelzl@40703
   367
lemma Cantor_Bernstein_aux:
hoelzl@40703
   368
  shows "\<exists>A' h. A' \<le> A \<and>
hoelzl@40703
   369
                (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
hoelzl@40703
   370
                (\<forall>a \<in> A'. h a = f a) \<and>
hoelzl@40703
   371
                (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
hoelzl@40703
   372
proof-
hoelzl@40703
   373
  obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
hoelzl@40703
   374
  have 0: "mono H" unfolding mono_def H_def by blast
hoelzl@40703
   375
  then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
hoelzl@40703
   376
  hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
hoelzl@40703
   377
  hence 3: "A' \<le> A" by blast
hoelzl@40703
   378
  have 4: "\<forall>a \<in> A'.  a \<notin> g`(B - f ` A')"
hoelzl@40703
   379
  using 2 by blast
hoelzl@40703
   380
  have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
hoelzl@40703
   381
  using 2 by blast
hoelzl@40703
   382
  (*  *)
hoelzl@40703
   383
  obtain h where h_def:
hoelzl@40703
   384
  "h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
hoelzl@40703
   385
  hence "\<forall>a \<in> A'. h a = f a" by auto
hoelzl@40703
   386
  moreover
hoelzl@40703
   387
  have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
hoelzl@40703
   388
  proof
hoelzl@40703
   389
    fix a assume *: "a \<in> A - A'"
hoelzl@40703
   390
    let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
hoelzl@40703
   391
    have "h a = (SOME b. ?phi b)" using h_def * by auto
hoelzl@40703
   392
    moreover have "\<exists>b. ?phi b" using 5 *  by auto
hoelzl@40703
   393
    ultimately show  "?phi (h a)" using someI_ex[of ?phi] by auto
hoelzl@40703
   394
  qed
hoelzl@40703
   395
  ultimately show ?thesis using 3 4 by blast
hoelzl@40703
   396
qed
hoelzl@40703
   397
hoelzl@40703
   398
theorem Cantor_Bernstein:
hoelzl@40703
   399
  assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
hoelzl@40703
   400
          INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
hoelzl@40703
   401
  shows "\<exists>h. bij_betw h A B"
hoelzl@40703
   402
proof-
hoelzl@40703
   403
  obtain A' and h where 0: "A' \<le> A" and
hoelzl@40703
   404
  1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
hoelzl@40703
   405
  2: "\<forall>a \<in> A'. h a = f a" and
hoelzl@40703
   406
  3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
hoelzl@40703
   407
  using Cantor_Bernstein_aux[of A g B f] by blast
hoelzl@40703
   408
  have "inj_on h A"
hoelzl@40703
   409
  proof (intro inj_onI)
hoelzl@40703
   410
    fix a1 a2
hoelzl@40703
   411
    assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
hoelzl@40703
   412
    show "a1 = a2"
hoelzl@40703
   413
    proof(cases "a1 \<in> A'")
hoelzl@40703
   414
      assume Case1: "a1 \<in> A'"
hoelzl@40703
   415
      show ?thesis
hoelzl@40703
   416
      proof(cases "a2 \<in> A'")
hoelzl@40703
   417
        assume Case11: "a2 \<in> A'"
hoelzl@40703
   418
        hence "f a1 = f a2" using Case1 2 6 by auto
hoelzl@40703
   419
        thus ?thesis using INJ1 Case1 Case11 0
hoelzl@40703
   420
        unfolding inj_on_def by blast
hoelzl@40703
   421
      next
hoelzl@40703
   422
        assume Case12: "a2 \<notin> A'"
hoelzl@40703
   423
        hence False using 3 5 2 6 Case1 by force
hoelzl@40703
   424
        thus ?thesis by simp
hoelzl@40703
   425
      qed
hoelzl@40703
   426
    next
hoelzl@40703
   427
    assume Case2: "a1 \<notin> A'"
hoelzl@40703
   428
      show ?thesis
hoelzl@40703
   429
      proof(cases "a2 \<in> A'")
hoelzl@40703
   430
        assume Case21: "a2 \<in> A'"
hoelzl@40703
   431
        hence False using 3 4 2 6 Case2 by auto
hoelzl@40703
   432
        thus ?thesis by simp
hoelzl@40703
   433
      next
hoelzl@40703
   434
        assume Case22: "a2 \<notin> A'"
hoelzl@40703
   435
        hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
hoelzl@40703
   436
        thus ?thesis using 6 by simp
hoelzl@40703
   437
      qed
hoelzl@40703
   438
    qed
hoelzl@40703
   439
  qed
hoelzl@40703
   440
  (*  *)
hoelzl@40703
   441
  moreover
hoelzl@40703
   442
  have "h ` A = B"
hoelzl@40703
   443
  proof safe
hoelzl@40703
   444
    fix a assume "a \<in> A"
hoelzl@40703
   445
    thus "h a \<in> B" using SUB1 2 3 by (case_tac "a \<in> A'", auto)
hoelzl@40703
   446
  next
hoelzl@40703
   447
    fix b assume *: "b \<in> B"
hoelzl@40703
   448
    show "b \<in> h ` A"
hoelzl@40703
   449
    proof(cases "b \<in> f ` A'")
hoelzl@40703
   450
      assume Case1: "b \<in> f ` A'"
hoelzl@40703
   451
      then obtain a where "a \<in> A' \<and> b = f a" by blast
hoelzl@40703
   452
      thus ?thesis using 2 0 by force
hoelzl@40703
   453
    next
hoelzl@40703
   454
      assume Case2: "b \<notin> f ` A'"
hoelzl@40703
   455
      hence "g b \<notin> A'" using 1 * by auto
hoelzl@40703
   456
      hence 4: "g b \<in> A - A'" using * SUB2 by auto
hoelzl@40703
   457
      hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
hoelzl@40703
   458
      using 3 by auto
hoelzl@40703
   459
      hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
hoelzl@40703
   460
      thus ?thesis using 4 by force
hoelzl@40703
   461
    qed
hoelzl@40703
   462
  qed
hoelzl@40703
   463
  (*  *)
hoelzl@40703
   464
  ultimately show ?thesis unfolding bij_betw_def by auto
hoelzl@40703
   465
qed
paulson@14760
   466
paulson@14760
   467
subsection {*Other Consequences of Hilbert's Epsilon*}
paulson@14760
   468
paulson@14760
   469
text {*Hilbert's Epsilon and the @{term split} Operator*}
paulson@14760
   470
paulson@14760
   471
text{*Looping simprule*}
paulson@14760
   472
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
haftmann@26347
   473
  by simp
paulson@14760
   474
paulson@14760
   475
lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
haftmann@26347
   476
  by (simp add: split_def)
paulson@14760
   477
paulson@14760
   478
lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
haftmann@26347
   479
  by blast
paulson@14760
   480
paulson@14760
   481
paulson@14760
   482
text{*A relation is wellfounded iff it has no infinite descending chain*}
paulson@14760
   483
lemma wf_iff_no_infinite_down_chain:
paulson@14760
   484
  "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
paulson@14760
   485
apply (simp only: wf_eq_minimal)
paulson@14760
   486
apply (rule iffI)
paulson@14760
   487
 apply (rule notI)
paulson@14760
   488
 apply (erule exE)
paulson@14760
   489
 apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
paulson@14760
   490
apply (erule contrapos_np, simp, clarify)
paulson@14760
   491
apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
paulson@14760
   492
 apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
paulson@14760
   493
 apply (rule allI, simp)
paulson@14760
   494
 apply (rule someI2_ex, blast, blast)
paulson@14760
   495
apply (rule allI)
paulson@14760
   496
apply (induct_tac "n", simp_all)
paulson@14760
   497
apply (rule someI2_ex, blast+)
paulson@14760
   498
done
paulson@14760
   499
nipkow@27760
   500
lemma wf_no_infinite_down_chainE:
nipkow@27760
   501
  assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
nipkow@27760
   502
using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
nipkow@27760
   503
nipkow@27760
   504
paulson@14760
   505
text{*A dynamically-scoped fact for TFL *}
wenzelm@12298
   506
lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
wenzelm@12298
   507
  by (blast intro: someI)
paulson@11451
   508
wenzelm@12298
   509
wenzelm@12298
   510
subsection {* Least value operator *}
paulson@11451
   511
haftmann@35416
   512
definition
haftmann@35416
   513
  LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
paulson@14760
   514
  "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
paulson@11451
   515
paulson@11451
   516
syntax
wenzelm@12298
   517
  "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   518
translations
wenzelm@35115
   519
  "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
paulson@11451
   520
paulson@11451
   521
lemma LeastMI2:
wenzelm@12298
   522
  "P x ==> (!!y. P y ==> m x <= m y)
wenzelm@12298
   523
    ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
wenzelm@12298
   524
    ==> Q (LeastM m P)"
paulson@14760
   525
  apply (simp add: LeastM_def)
paulson@14208
   526
  apply (rule someI2_ex, blast, blast)
wenzelm@12298
   527
  done
paulson@11451
   528
paulson@11451
   529
lemma LeastM_equality:
wenzelm@12298
   530
  "P k ==> (!!x. P x ==> m k <= m x)
wenzelm@12298
   531
    ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
paulson@14208
   532
  apply (rule LeastMI2, assumption, blast)
wenzelm@12298
   533
  apply (blast intro!: order_antisym)
wenzelm@12298
   534
  done
paulson@11451
   535
paulson@11454
   536
lemma wf_linord_ex_has_least:
paulson@14760
   537
  "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
paulson@14760
   538
    ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
wenzelm@12298
   539
  apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
paulson@14208
   540
  apply (drule_tac x = "m`Collect P" in spec, force)
wenzelm@12298
   541
  done
paulson@11454
   542
paulson@11454
   543
lemma ex_has_least_nat:
paulson@14760
   544
    "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
wenzelm@12298
   545
  apply (simp only: pred_nat_trancl_eq_le [symmetric])
wenzelm@12298
   546
  apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
paulson@16796
   547
   apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
wenzelm@12298
   548
  done
paulson@11454
   549
wenzelm@12298
   550
lemma LeastM_nat_lemma:
paulson@14760
   551
    "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
paulson@14760
   552
  apply (simp add: LeastM_def)
wenzelm@12298
   553
  apply (rule someI_ex)
wenzelm@12298
   554
  apply (erule ex_has_least_nat)
wenzelm@12298
   555
  done
paulson@11454
   556
paulson@11454
   557
lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
paulson@11454
   558
paulson@11454
   559
lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
paulson@14208
   560
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
paulson@11454
   561
paulson@11451
   562
wenzelm@12298
   563
subsection {* Greatest value operator *}
paulson@11451
   564
haftmann@35416
   565
definition
haftmann@35416
   566
  GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
paulson@14760
   567
  "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
wenzelm@12298
   568
haftmann@35416
   569
definition
haftmann@35416
   570
  Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
wenzelm@12298
   571
  "Greatest == GreatestM (%x. x)"
paulson@11451
   572
paulson@11451
   573
syntax
wenzelm@35115
   574
  "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
wenzelm@12298
   575
      ("GREATEST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   576
translations
wenzelm@35115
   577
  "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
paulson@11451
   578
paulson@11451
   579
lemma GreatestMI2:
wenzelm@12298
   580
  "P x ==> (!!y. P y ==> m y <= m x)
wenzelm@12298
   581
    ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
wenzelm@12298
   582
    ==> Q (GreatestM m P)"
paulson@14760
   583
  apply (simp add: GreatestM_def)
paulson@14208
   584
  apply (rule someI2_ex, blast, blast)
wenzelm@12298
   585
  done
paulson@11451
   586
paulson@11451
   587
lemma GreatestM_equality:
wenzelm@12298
   588
 "P k ==> (!!x. P x ==> m x <= m k)
wenzelm@12298
   589
    ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
paulson@14208
   590
  apply (rule_tac m = m in GreatestMI2, assumption, blast)
wenzelm@12298
   591
  apply (blast intro!: order_antisym)
wenzelm@12298
   592
  done
paulson@11451
   593
paulson@11451
   594
lemma Greatest_equality:
wenzelm@12298
   595
  "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
paulson@14760
   596
  apply (simp add: Greatest_def)
paulson@14208
   597
  apply (erule GreatestM_equality, blast)
wenzelm@12298
   598
  done
paulson@11451
   599
paulson@11451
   600
lemma ex_has_greatest_nat_lemma:
paulson@14760
   601
  "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
paulson@14760
   602
    ==> \<exists>y. P y & ~ (m y < m k + n)"
paulson@15251
   603
  apply (induct n, force)
wenzelm@12298
   604
  apply (force simp add: le_Suc_eq)
wenzelm@12298
   605
  done
paulson@11451
   606
wenzelm@12298
   607
lemma ex_has_greatest_nat:
paulson@14760
   608
  "P k ==> \<forall>y. P y --> m y < b
paulson@14760
   609
    ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
wenzelm@12298
   610
  apply (rule ccontr)
wenzelm@12298
   611
  apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
paulson@14208
   612
    apply (subgoal_tac [3] "m k <= b", auto)
wenzelm@12298
   613
  done
paulson@11451
   614
wenzelm@12298
   615
lemma GreatestM_nat_lemma:
paulson@14760
   616
  "P k ==> \<forall>y. P y --> m y < b
paulson@14760
   617
    ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
paulson@14760
   618
  apply (simp add: GreatestM_def)
wenzelm@12298
   619
  apply (rule someI_ex)
paulson@14208
   620
  apply (erule ex_has_greatest_nat, assumption)
wenzelm@12298
   621
  done
paulson@11451
   622
paulson@11451
   623
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
paulson@11451
   624
wenzelm@12298
   625
lemma GreatestM_nat_le:
paulson@14760
   626
  "P x ==> \<forall>y. P y --> m y < b
wenzelm@12298
   627
    ==> (m x::nat) <= m (GreatestM m P)"
berghofe@21020
   628
  apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
wenzelm@12298
   629
  done
wenzelm@12298
   630
wenzelm@12298
   631
wenzelm@12298
   632
text {* \medskip Specialization to @{text GREATEST}. *}
wenzelm@12298
   633
paulson@14760
   634
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
paulson@14760
   635
  apply (simp add: Greatest_def)
paulson@14208
   636
  apply (rule GreatestM_natI, auto)
wenzelm@12298
   637
  done
paulson@11451
   638
wenzelm@12298
   639
lemma Greatest_le:
paulson@14760
   640
    "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
paulson@14760
   641
  apply (simp add: Greatest_def)
paulson@14208
   642
  apply (rule GreatestM_nat_le, auto)
wenzelm@12298
   643
  done
wenzelm@12298
   644
wenzelm@12298
   645
wenzelm@17893
   646
subsection {* Specification package -- Hilbertized version *}
wenzelm@17893
   647
wenzelm@17893
   648
lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
wenzelm@17893
   649
  by (simp only: someI_ex)
wenzelm@17893
   650
haftmann@31723
   651
use "Tools/choice_specification.ML"
skalberg@14115
   652
paulson@11451
   653
end