src/HOL/Hilbert_Choice.thy
author nipkow
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(*  Title:      HOL/Hilbert_Choice.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson
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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
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uses ("Tools/meson.ML") ("Tools/specification_package.ML")
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begin
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subsection {* Hilbert's epsilon *}
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axiomatization
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  Eps :: "('a => bool) => 'a"
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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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(* to avoid eta-contraction of body *)
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[(@{const_syntax Eps}, fn [Abs abs] =>
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     let val (x,t) = atomic_abs_tr' abs
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     in Syntax.const "_Eps" $ x $ t end)]
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*}
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constdefs
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  inv :: "('a => 'b) => ('b => 'a)"
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  "inv(f :: 'a => 'b) == %y. SOME x. f x = y"
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  Inv :: "'a set => ('a => 'b) => ('b => 'a)"
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  "Inv A f == %x. SOME y. y \<in> A & f y = x"
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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 intro: some_equality)
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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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text{*Used in @{text "Tools/meson.ML"}*}
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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_id [simp]: "inv id = id"
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by (simp add: inv_def id_def)
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text{*A one-to-one function has an inverse.*}
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lemma inv_f_f [simp]: "inj f ==> inv f (f x) = x"
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by (simp add: inv_def inj_eq)
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lemma inv_f_eq: "[| inj f;  f x = y |] ==> inv f y = x"
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apply (erule subst)
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apply (erule inv_f_f)
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done
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lemma inj_imp_inv_eq: "[| inj f; \<forall>x. f(g x) = x |] ==> inv f = g"
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by (blast intro: ext inv_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_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 expand_fun_eq)
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apply (blast intro: inj_on_inverseI inv_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_image_cancel[simp]:
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  "inj f ==> inv f ` f ` S = S"
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by (simp add: image_compose[symmetric])
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lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
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by (blast intro: surjI inv_f_f)
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lemma f_inv_f: "y \<in> range(f) ==> f(inv f y) = y"
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apply (simp add: inv_def)
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apply (fast intro: someI)
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done
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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_f surj_range)
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lemma inv_injective:
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  assumes eq: "inv f x = inv f y"
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      and x: "x: range f"
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      and y: "y: range f"
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  shows "x=y"
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proof -
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  have "f (inv f x) = f (inv f y)" using eq by simp
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  thus ?thesis by (simp add: f_inv_f x y) 
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qed
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lemma inj_on_inv: "A <= range(f) ==> inj_on (inv f) A"
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by (fast intro: inj_onI elim: inv_injective injD)
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lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
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by (simp add: inj_on_inv surj_range)
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lemma surj_iff: "(surj f) = (f o inv f = id)"
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apply (simp add: o_def expand_fun_eq)
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apply (blast intro: surjI surj_f_inv_f)
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done
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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_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 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_f_f, symmetric])
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done
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subsection {*Inverse of a PI-function (restricted domain)*}
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lemma Inv_f_f: "[| inj_on f A;  x \<in> A |] ==> Inv A f (f x) = x"
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apply (simp add: Inv_def inj_on_def)
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apply (blast intro: someI2)
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done
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lemma f_Inv_f: "y \<in> f`A  ==> f (Inv A f y) = y"
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apply (simp add: Inv_def)
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apply (fast intro: someI2)
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done
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lemma Inv_injective:
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  assumes eq: "Inv A f x = Inv 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 A f x) = f (Inv A f y)" using eq by simp
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  thus ?thesis by (simp add: f_Inv_f x y) 
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qed
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lemma inj_on_Inv: "B <= f`A ==> inj_on (Inv A f) B"
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apply (rule inj_onI)
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apply (blast intro: inj_onI dest: Inv_injective injD)
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done
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lemma Inv_mem: "[| f ` A = B;  x \<in> B |] ==> Inv A f x \<in> A"
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apply (simp add: Inv_def)
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apply (fast intro: someI2)
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done
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lemma Inv_f_eq: "[| inj_on f A; f x = y; x \<in> A |] ==> Inv A f y = x"
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  apply (erule subst)
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  apply (erule Inv_f_f, assumption)
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  done
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lemma Inv_comp:
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  "[| inj_on f (g ` A); inj_on g A; x \<in> f ` g ` A |] ==>
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  Inv A (f o g) x = (Inv A g o Inv (g ` A) f) x"
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  apply simp
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  apply (rule Inv_f_eq)
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    apply (fast intro: comp_inj_on)
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   apply (simp add: f_Inv_f Inv_mem)
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  apply (simp add: Inv_mem)
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  done
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lemma bij_betw_Inv: "bij_betw f A B \<Longrightarrow> bij_betw (Inv A f) B A"
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  apply (auto simp add: bij_betw_def inj_on_Inv Inv_mem)
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  apply (simp add: image_compose [symmetric] o_def)
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  apply (simp add: image_def Inv_f_f)
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  done
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subsection {*Other Consequences of Hilbert's Epsilon*}
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text {*Hilbert's Epsilon and the @{term split} Operator*}
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text{*Looping simprule*}
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lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
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  by simp
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lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
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  by (simp add: split_def)
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lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
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  by blast
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text{*A relation is wellfounded iff it has no infinite descending chain*}
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lemma wf_iff_no_infinite_down_chain:
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  "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
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apply (simp only: wf_eq_minimal)
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apply (rule iffI)
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 apply (rule notI)
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 apply (erule exE)
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 apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
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apply (erule contrapos_np, simp, clarify)
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apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
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 apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
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 apply (rule allI, simp)
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 apply (rule someI2_ex, blast, blast)
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apply (rule allI)
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apply (induct_tac "n", simp_all)
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apply (rule someI2_ex, blast+)
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done
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lemma wf_no_infinite_down_chainE:
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  assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
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using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
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3aa86edac080 added lemma
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text{*A dynamically-scoped fact for TFL *}
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lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
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  by (blast intro: someI)
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subsection {* Least value operator *}
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constdefs
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  LeastM :: "['a => 'b::ord, 'a => bool] => 'a"
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  "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
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syntax
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  "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
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translations
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  "LEAST x WRT m. P" == "LeastM m (%x. P)"
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8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
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lemma LeastMI2:
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  "P x ==> (!!y. P y ==> m x <= m y)
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    ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
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    ==> Q (LeastM m P)"
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  apply (simp add: LeastM_def)
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  apply (rule someI2_ex, blast, blast)
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  done
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lemma LeastM_equality:
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  "P k ==> (!!x. P x ==> m k <= m x)
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    ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
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  apply (rule LeastMI2, assumption, blast)
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  apply (blast intro!: order_antisym)
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  done
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lemma wf_linord_ex_has_least:
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  "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
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    ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
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  apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
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  apply (drule_tac x = "m`Collect P" in spec, force)
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  done
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lemma ex_has_least_nat:
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    "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
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  apply (simp only: pred_nat_trancl_eq_le [symmetric])
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  apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
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   apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
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   358
  done
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   359
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lemma LeastM_nat_lemma:
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    "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
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  apply (simp add: LeastM_def)
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   363
  apply (rule someI_ex)
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   364
  apply (erule ex_has_least_nat)
wenzelm
parents: 12023
diff changeset
   365
  done
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   366
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lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
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   368
7514e5e21cb8 Hilbert restructuring: Wellfounded_Relations no longer needs Hilbert_Choice
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lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
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by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
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diff changeset
   371
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subsection {* Greatest value operator *}
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constdefs
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  GreatestM :: "['a => 'b::ord, 'a => bool] => 'a"
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  "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
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   379
  Greatest :: "('a::ord => bool) => 'a"    (binder "GREATEST " 10)
wenzelm
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   380
  "Greatest == GreatestM (%x. x)"
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   382
syntax
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   383
  "_GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"
wenzelm
parents: 12023
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   384
      ("GREATEST _ WRT _. _" [0, 4, 10] 10)
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   385
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
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   386
translations
12298
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diff changeset
   387
  "GREATEST x WRT m. P" == "GreatestM m (%x. P)"
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   388
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
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lemma GreatestMI2:
12298
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   390
  "P x ==> (!!y. P y ==> m y <= m x)
wenzelm
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   391
    ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
wenzelm
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   392
    ==> Q (GreatestM m P)"
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   393
  apply (simp add: GreatestM_def)
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   394
  apply (rule someI2_ex, blast, blast)
12298
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parents: 12023
diff changeset
   395
  done
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   396
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
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   397
lemma GreatestM_equality:
12298
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   398
 "P k ==> (!!x. P x ==> m x <= m k)
wenzelm
parents: 12023
diff changeset
   399
    ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
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paulson
parents: 14115
diff changeset
   400
  apply (rule_tac m = m in GreatestMI2, assumption, blast)
12298
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parents: 12023
diff changeset
   401
  apply (blast intro!: order_antisym)
wenzelm
parents: 12023
diff changeset
   402
  done
11451
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paulson
parents:
diff changeset
   403
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   404
lemma Greatest_equality:
12298
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diff changeset
   405
  "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
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diff changeset
   406
  apply (simp add: Greatest_def)
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   407
  apply (erule GreatestM_equality, blast)
12298
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parents: 12023
diff changeset
   408
  done
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paulson
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   409
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   410
lemma ex_has_greatest_nat_lemma:
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diff changeset
   411
  "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
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   412
    ==> \<exists>y. P y & ~ (m y < m k + n)"
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paulson
parents: 15140
diff changeset
   413
  apply (induct n, force)
12298
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diff changeset
   414
  apply (force simp add: le_Suc_eq)
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parents: 12023
diff changeset
   415
  done
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paulson
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diff changeset
   416
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diff changeset
   417
lemma ex_has_greatest_nat:
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   418
  "P k ==> \<forall>y. P y --> m y < b
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   419
    ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
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diff changeset
   420
  apply (rule ccontr)
wenzelm
parents: 12023
diff changeset
   421
  apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
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parents: 14115
diff changeset
   422
    apply (subgoal_tac [3] "m k <= b", auto)
12298
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diff changeset
   423
  done
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   424
12298
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   425
lemma GreatestM_nat_lemma:
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   426
  "P k ==> \<forall>y. P y --> m y < b
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paulson
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   427
    ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
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paulson
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diff changeset
   428
  apply (simp add: GreatestM_def)
12298
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diff changeset
   429
  apply (rule someI_ex)
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parents: 14115
diff changeset
   430
  apply (erule ex_has_greatest_nat, assumption)
12298
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diff changeset
   431
  done
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paulson
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   432
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
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diff changeset
   433
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   434
12298
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diff changeset
   435
lemma GreatestM_nat_le:
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paulson
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diff changeset
   436
  "P x ==> \<forall>y. P y --> m y < b
12298
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diff changeset
   437
    ==> (m x::nat) <= m (GreatestM m P)"
21020
9af9ceb16d58 Adapted to changes in FixedPoint theory.
berghofe
parents: 18389
diff changeset
   438
  apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
12298
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parents: 12023
diff changeset
   439
  done
wenzelm
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diff changeset
   440
wenzelm
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diff changeset
   441
wenzelm
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diff changeset
   442
text {* \medskip Specialization to @{text GREATEST}. *}
wenzelm
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diff changeset
   443
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paulson
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diff changeset
   444
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
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paulson
parents: 14399
diff changeset
   445
  apply (simp add: Greatest_def)
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paulson
parents: 14115
diff changeset
   446
  apply (rule GreatestM_natI, auto)
12298
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diff changeset
   447
  done
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paulson
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diff changeset
   448
12298
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diff changeset
   449
lemma Greatest_le:
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diff changeset
   450
    "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
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paulson
parents: 14399
diff changeset
   451
  apply (simp add: Greatest_def)
14208
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paulson
parents: 14115
diff changeset
   452
  apply (rule GreatestM_nat_le, auto)
12298
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parents: 12023
diff changeset
   453
  done
wenzelm
parents: 12023
diff changeset
   454
wenzelm
parents: 12023
diff changeset
   455
wenzelm
parents: 12023
diff changeset
   456
subsection {* The Meson proof procedure *}
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paulson
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   457
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diff changeset
   458
subsubsection {* Negation Normal Form *}
wenzelm
parents: 12023
diff changeset
   459
wenzelm
parents: 12023
diff changeset
   460
text {* de Morgan laws *}
wenzelm
parents: 12023
diff changeset
   461
wenzelm
parents: 12023
diff changeset
   462
lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
wenzelm
parents: 12023
diff changeset
   463
  and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
wenzelm
parents: 12023
diff changeset
   464
  and meson_not_notD: "~~P ==> P"
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paulson
parents: 14399
diff changeset
   465
  and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   466
  and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
12298
wenzelm
parents: 12023
diff changeset
   467
  by fast+
11451
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paulson
parents:
diff changeset
   468
12298
wenzelm
parents: 12023
diff changeset
   469
text {* Removal of @{text "-->"} and @{text "<->"} (positive and
wenzelm
parents: 12023
diff changeset
   470
negative occurrences) *}
wenzelm
parents: 12023
diff changeset
   471
wenzelm
parents: 12023
diff changeset
   472
lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
wenzelm
parents: 12023
diff changeset
   473
  and meson_not_impD: "~(P-->Q) ==> P & ~Q"
wenzelm
parents: 12023
diff changeset
   474
  and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
wenzelm
parents: 12023
diff changeset
   475
  and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
wenzelm
parents: 12023
diff changeset
   476
    -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
18389
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paulson
parents: 17893
diff changeset
   477
  and meson_not_refl_disj_D: "x ~= x | P ==> P"
12298
wenzelm
parents: 12023
diff changeset
   478
  by fast+
wenzelm
parents: 12023
diff changeset
   479
wenzelm
parents: 12023
diff changeset
   480
wenzelm
parents: 12023
diff changeset
   481
subsubsection {* Pulling out the existential quantifiers *}
wenzelm
parents: 12023
diff changeset
   482
wenzelm
parents: 12023
diff changeset
   483
text {* Conjunction *}
wenzelm
parents: 12023
diff changeset
   484
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a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   485
lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   486
  and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
12298
wenzelm
parents: 12023
diff changeset
   487
  by fast+
wenzelm
parents: 12023
diff changeset
   488
11451
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paulson
parents:
diff changeset
   489
12298
wenzelm
parents: 12023
diff changeset
   490
text {* Disjunction *}
wenzelm
parents: 12023
diff changeset
   491
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a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   492
lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
12298
wenzelm
parents: 12023
diff changeset
   493
  -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
wenzelm
parents: 12023
diff changeset
   494
  -- {* With ex-Skolemization, makes fewer Skolem constants *}
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   495
  and meson_disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   496
  and meson_disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
12298
wenzelm
parents: 12023
diff changeset
   497
  by fast+
wenzelm
parents: 12023
diff changeset
   498
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   499
12298
wenzelm
parents: 12023
diff changeset
   500
subsubsection {* Generating clauses for the Meson Proof Procedure *}
wenzelm
parents: 12023
diff changeset
   501
wenzelm
parents: 12023
diff changeset
   502
text {* Disjunctions *}
wenzelm
parents: 12023
diff changeset
   503
wenzelm
parents: 12023
diff changeset
   504
lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"
wenzelm
parents: 12023
diff changeset
   505
  and meson_disj_comm: "P|Q ==> Q|P"
wenzelm
parents: 12023
diff changeset
   506
  and meson_disj_FalseD1: "False|P ==> P"
wenzelm
parents: 12023
diff changeset
   507
  and meson_disj_FalseD2: "P|False ==> P"
wenzelm
parents: 12023
diff changeset
   508
  by fast+
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   509
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   510
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   511
subsection{*Lemmas for Meson, the Model Elimination Procedure*}
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
text{* Generation of contrapositives *}
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{*Inserts negated disjunct after removing the negation; P is a literal.
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   516
  Model elimination requires assuming the negation of every attempted subgoal,
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   517
  hence the negated disjuncts.*}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   518
lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   519
by blast
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
text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   522
lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   523
by blast
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   524
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   525
text{*@{term P} should be a literal*}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   526
lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   527
by blast
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   528
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   529
text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   530
insert new assumptions, for ordinary resolution.*}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   531
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   532
lemmas make_neg_rule' = make_refined_neg_rule
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   533
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   534
lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   535
by blast
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   536
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   537
text{* Generation of a goal clause -- put away the final literal *}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   538
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   539
lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   540
by blast
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   541
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   542
lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   543
by blast
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   544
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   545
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   546
subsubsection{* Lemmas for Forward Proof*}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   547
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   548
text{*There is a similarity to congruence rules*}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   549
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   550
(*NOTE: could handle conjunctions (faster?) by
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   551
    nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   552
lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   553
by blast
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   554
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   555
lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   556
by blast
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   557
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   558
(*Version of @{text disj_forward} for removal of duplicate literals*)
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   559
lemma disj_forward2:
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   560
    "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   561
apply blast 
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   562
done
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   563
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   564
lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   565
by blast
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   566
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   567
lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   568
by blast
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   569
17420
bdcffa8d8665 comment
paulson
parents: 16796
diff changeset
   570
bdcffa8d8665 comment
paulson
parents: 16796
diff changeset
   571
text{*Many of these bindings are used by the ATP linkup, and not just by
bdcffa8d8665 comment
paulson
parents: 16796
diff changeset
   572
legacy proof scripts.*}
14760
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   573
ML
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   574
{*
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   575
val inv_def = thm "inv_def";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   576
val Inv_def = thm "Inv_def";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   577
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   578
val someI = thm "someI";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   579
val someI_ex = thm "someI_ex";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   580
val someI2 = thm "someI2";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   581
val someI2_ex = thm "someI2_ex";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   582
val some_equality = thm "some_equality";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   583
val some1_equality = thm "some1_equality";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   584
val some_eq_ex = thm "some_eq_ex";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   585
val some_eq_trivial = thm "some_eq_trivial";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   586
val some_sym_eq_trivial = thm "some_sym_eq_trivial";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   587
val choice = thm "choice";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   588
val bchoice = thm "bchoice";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   589
val inv_id = thm "inv_id";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   590
val inv_f_f = thm "inv_f_f";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   591
val inv_f_eq = thm "inv_f_eq";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   592
val inj_imp_inv_eq = thm "inj_imp_inv_eq";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   593
val inj_transfer = thm "inj_transfer";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   594
val inj_iff = thm "inj_iff";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   595
val inj_imp_surj_inv = thm "inj_imp_surj_inv";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   596
val f_inv_f = thm "f_inv_f";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   597
val surj_f_inv_f = thm "surj_f_inv_f";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   598
val inv_injective = thm "inv_injective";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   599
val inj_on_inv = thm "inj_on_inv";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   600
val surj_imp_inj_inv = thm "surj_imp_inj_inv";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   601
val surj_iff = thm "surj_iff";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   602
val surj_imp_inv_eq = thm "surj_imp_inv_eq";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   603
val bij_imp_bij_inv = thm "bij_imp_bij_inv";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   604
val inv_equality = thm "inv_equality";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   605
val inv_inv_eq = thm "inv_inv_eq";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   606
val o_inv_distrib = thm "o_inv_distrib";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   607
val image_surj_f_inv_f = thm "image_surj_f_inv_f";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   608
val image_inv_f_f = thm "image_inv_f_f";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   609
val inv_image_comp = thm "inv_image_comp";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   610
val bij_image_Collect_eq = thm "bij_image_Collect_eq";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   611
val bij_vimage_eq_inv_image = thm "bij_vimage_eq_inv_image";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   612
val Inv_f_f = thm "Inv_f_f";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   613
val f_Inv_f = thm "f_Inv_f";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   614
val Inv_injective = thm "Inv_injective";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   615
val inj_on_Inv = thm "inj_on_Inv";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   616
val split_paired_Eps = thm "split_paired_Eps";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   617
val Eps_split = thm "Eps_split";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   618
val Eps_split_eq = thm "Eps_split_eq";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   619
val wf_iff_no_infinite_down_chain = thm "wf_iff_no_infinite_down_chain";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   620
val Inv_mem = thm "Inv_mem";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   621
val Inv_f_eq = thm "Inv_f_eq";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   622
val Inv_comp = thm "Inv_comp";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   623
val tfl_some = thm "tfl_some";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   624
val make_neg_rule = thm "make_neg_rule";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   625
val make_refined_neg_rule = thm "make_refined_neg_rule";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   626
val make_pos_rule = thm "make_pos_rule";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   627
val make_neg_rule' = thm "make_neg_rule'";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   628
val make_pos_rule' = thm "make_pos_rule'";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   629
val make_neg_goal = thm "make_neg_goal";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   630
val make_pos_goal = thm "make_pos_goal";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   631
val conj_forward = thm "conj_forward";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   632
val disj_forward = thm "disj_forward";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   633
val disj_forward2 = thm "disj_forward2";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   634
val all_forward = thm "all_forward";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   635
val ex_forward = thm "ex_forward";
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   636
*}
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   637
a08e916f4946 conversion of Hilbert_Choice to Isar script
paulson
parents: 14399
diff changeset
   638
21999
0cf192e489e2 improvements to proof reconstruction. Some files loaded in a different order
paulson
parents: 21243
diff changeset
   639
subsection {* Meson package *}
17893
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   640
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   641
use "Tools/meson.ML"
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents:
diff changeset
   642
26562
9d25ef112cf6 * Metis: the maximum number of clauses that can be produced from a theorem is now given by the attribute max_clauses. Theorems that exceed this number are ignored, with a warning printed.
paulson
parents: 26347
diff changeset
   643
setup Meson.setup
9d25ef112cf6 * Metis: the maximum number of clauses that can be produced from a theorem is now given by the attribute max_clauses. Theorems that exceed this number are ignored, with a warning printed.
paulson
parents: 26347
diff changeset
   644
17893
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   645
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   646
subsection {* Specification package -- Hilbertized version *}
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   647
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   648
lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   649
  by (simp only: someI_ex)
aef5a6d11c2a added lemma exE_some (from specification_package.ML);
wenzelm
parents: 17702
diff changeset
   650
14115
65ec3f73d00b Added package for definition by specification.
skalberg
parents: 13764
diff changeset
   651
use "Tools/specification_package.ML"
65ec3f73d00b Added package for definition by specification.
skalberg
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11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
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end