more qualified names -- eliminated hide_const (open);
(* Title: HOL/Fun.thy Author: Tobias Nipkow, Cambridge University Computer Laboratory Author: Andrei Popescu, TU Muenchen Copyright 1994, 2012*)section {* Notions about functions *}theory Funimports Setkeywords "functor" :: thy_goalbeginlemma apply_inverse: "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u" by autotext{*Uniqueness, so NOT the axiom of choice.*}lemma uniq_choice: "\<forall>x. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)" by (force intro: theI')lemma b_uniq_choice: "\<forall>x\<in>S. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x\<in>S. Q x (f x)" by (force intro: theI')subsection {* The Identity Function @{text id} *}definition id :: "'a \<Rightarrow> 'a" where "id = (\<lambda>x. x)"lemma id_apply [simp]: "id x = x" by (simp add: id_def)lemma image_id [simp]: "image id = id" by (simp add: id_def fun_eq_iff)lemma vimage_id [simp]: "vimage id = id" by (simp add: id_def fun_eq_iff)code_printing constant id \<rightharpoonup> (Haskell) "id"subsection {* The Composition Operator @{text "f \<circ> g"} *}definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55) where "f o g = (\<lambda>x. f (g x))"notation (xsymbols) comp (infixl "\<circ>" 55)notation (HTML output) comp (infixl "\<circ>" 55)lemma comp_apply [simp]: "(f o g) x = f (g x)" by (simp add: comp_def)lemma comp_assoc: "(f o g) o h = f o (g o h)" by (simp add: fun_eq_iff)lemma id_comp [simp]: "id o g = g" by (simp add: fun_eq_iff)lemma comp_id [simp]: "f o id = f" by (simp add: fun_eq_iff)lemma comp_eq_dest: "a o b = c o d \<Longrightarrow> a (b v) = c (d v)" by (simp add: fun_eq_iff)lemma comp_eq_elim: "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R" by (simp add: fun_eq_iff) lemma comp_eq_dest_lhs: "a o b = c \<Longrightarrow> a (b v) = c v" by clarsimplemma comp_eq_id_dest: "a o b = id o c \<Longrightarrow> a (b v) = c v" by clarsimplemma image_comp: "f ` (g ` r) = (f o g) ` r" by autolemma vimage_comp: "f -` (g -` x) = (g \<circ> f) -` x" by autolemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h o f) ` A = (h o g) ` B" by (auto simp: comp_def elim!: equalityE)lemma image_bind: "f ` (Set.bind A g) = Set.bind A (op ` f \<circ> g)"by(auto simp add: Set.bind_def)lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g \<circ> f)"by(auto simp add: Set.bind_def)code_printing constant comp \<rightharpoonup> (SML) infixl 5 "o" and (Haskell) infixr 9 "."subsection {* The Forward Composition Operator @{text fcomp} *}definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) where "f \<circ>> g = (\<lambda>x. g (f x))"lemma fcomp_apply [simp]: "(f \<circ>> g) x = g (f x)" by (simp add: fcomp_def)lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)" by (simp add: fcomp_def)lemma id_fcomp [simp]: "id \<circ>> g = g" by (simp add: fcomp_def)lemma fcomp_id [simp]: "f \<circ>> id = f" by (simp add: fcomp_def)code_printing constant fcomp \<rightharpoonup> (Eval) infixl 1 "#>"no_notation fcomp (infixl "\<circ>>" 60)subsection {* Mapping functions *}definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" where "map_fun f g h = g \<circ> h \<circ> f"lemma map_fun_apply [simp]: "map_fun f g h x = g (h (f x))" by (simp add: map_fun_def)subsection {* Injectivity and Bijectivity *}definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" where -- "injective" "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where -- "bijective" "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"text{*A common special case: functions injective, surjective or bijective overthe entire domain type.*}abbreviation "inj f \<equiv> inj_on f UNIV"abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" where -- "surjective" "surj f \<equiv> (range f = UNIV)"abbreviation "bij f \<equiv> bij_betw f UNIV UNIV"text{* The negated case: *}translations"\<not> CONST surj f" <= "CONST range f \<noteq> CONST UNIV"lemma injI: assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "inj f" using assms unfolding inj_on_def by autotheorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)" by (unfold inj_on_def, blast)lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"by (simp add: inj_on_def)lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"by (force simp add: inj_on_def)lemma inj_on_cong: "(\<And> a. a : A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A = inj_on g A"unfolding inj_on_def by autolemma inj_on_strict_subset: "inj_on f B \<Longrightarrow> A \<subset> B \<Longrightarrow> f ` A \<subset> f ` B" unfolding inj_on_def by blastlemma inj_comp: "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)" by (simp add: inj_on_def)lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)" by (simp add: inj_on_def fun_eq_iff)lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"by (simp add: inj_on_eq_iff)lemma inj_on_id[simp]: "inj_on id A" by (simp add: inj_on_def)lemma inj_on_id2[simp]: "inj_on (%x. x) A"by (simp add: inj_on_def)lemma inj_on_Int: "inj_on f A \<or> inj_on f B \<Longrightarrow> inj_on f (A \<inter> B)"unfolding inj_on_def by blastlemma surj_id: "surj id"by simplemma bij_id[simp]: "bij id"by (simp add: bij_betw_def)lemma inj_onI: "(!! x y. [| x:A; y:A; f(x) = f(y) |] ==> x=y) ==> inj_on f A"by (simp add: inj_on_def)lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"by (auto dest: arg_cong [of concl: g] simp add: inj_on_def)lemma inj_onD: "[| inj_on f A; f(x)=f(y); x:A; y:A |] ==> x=y"by (unfold inj_on_def, blast)lemma inj_on_iff: "[| inj_on f A; x:A; y:A |] ==> (f(x)=f(y)) = (x=y)" by (fact inj_on_eq_iff)lemma comp_inj_on: "[| inj_on f A; inj_on g (f`A) |] ==> inj_on (g o f) A"by (simp add: comp_def inj_on_def)lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)" by (simp add: inj_on_def) blastlemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y); inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"apply(unfold inj_on_def)apply blastdonelemma inj_on_contraD: "[| inj_on f A; ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)"by (unfold inj_on_def, blast)lemma inj_singleton: "inj (%s. {s})"by (simp add: inj_on_def)lemma inj_on_empty[iff]: "inj_on f {}"by(simp add: inj_on_def)lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"by (unfold inj_on_def, blast)lemma inj_on_Un: "inj_on f (A Un B) = (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"apply(unfold inj_on_def)apply (blast intro:sym)donelemma inj_on_insert[iff]: "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"apply(unfold inj_on_def)apply (blast intro:sym)donelemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"apply(unfold inj_on_def)apply (blast)donelemma comp_inj_on_iff: "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A"by(auto simp add: comp_inj_on inj_on_def)lemma inj_on_imageI2: "inj_on (f' o f) A \<Longrightarrow> inj_on f A"by(auto simp add: comp_inj_on inj_on_def)lemma inj_img_insertE: assumes "inj_on f A" assumes "x \<notin> B" and "insert x B = f ` A" obtains x' A' where "x' \<notin> A'" and "A = insert x' A'" and "x = f x'" and "B = f ` A'"proof - from assms have "x \<in> f ` A" by auto then obtain x' where *: "x' \<in> A" "x = f x'" by auto then have "A = insert x' (A - {x'})" by auto with assms * have "B = f ` (A - {x'})" by (auto dest: inj_on_contraD) have "x' \<notin> A - {x'}" by simp from `x' \<notin> A - {x'}` `A = insert x' (A - {x'})` `x = f x'` `B = image f (A - {x'})` show ?thesis ..qedlemma linorder_injI: assumes hyp: "\<And>x y. x < (y::'a::linorder) \<Longrightarrow> f x \<noteq> f y" shows "inj f" -- {* Courtesy of Stephan Merz *}proof (rule inj_onI) fix x y assume f_eq: "f x = f y" show "x = y" by (rule linorder_cases) (auto dest: hyp simp: f_eq)qedlemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)" by autolemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g" using *[symmetric] by autolemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x" by (simp add: surj_def)lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C" by (simp add: surj_def, blast)lemma comp_surj: "[| surj f; surj g |] ==> surj (g o f)"apply (simp add: comp_def surj_def, clarify)apply (drule_tac x = y in spec, clarify)apply (drule_tac x = x in spec, blast)donelemma bij_betw_imageI: "\<lbrakk> inj_on f A; f ` A = B \<rbrakk> \<Longrightarrow> bij_betw f A B"unfolding bij_betw_def by clarifylemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> f ` A = B" unfolding bij_betw_def by clarifylemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f" unfolding bij_betw_def by autolemma bij_betw_empty1: assumes "bij_betw f {} A" shows "A = {}"using assms unfolding bij_betw_def by blastlemma bij_betw_empty2: assumes "bij_betw f A {}" shows "A = {}"using assms unfolding bij_betw_def by blastlemma inj_on_imp_bij_betw: "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"unfolding bij_betw_def by simplemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f" unfolding bij_betw_def ..lemma bijI: "[| inj f; surj f |] ==> bij f"by (simp add: bij_def)lemma bij_is_inj: "bij f ==> inj f"by (simp add: bij_def)lemma bij_is_surj: "bij f ==> surj f"by (simp add: bij_def)lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"by (simp add: bij_betw_def)lemma bij_betw_trans: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"by(auto simp add:bij_betw_def comp_inj_on)lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)" by (rule bij_betw_trans)lemma bij_betw_comp_iff: "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"by(auto simp add: bij_betw_def inj_on_def)lemma bij_betw_comp_iff2: assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'" shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"using assmsproof(auto simp add: bij_betw_comp_iff) assume *: "bij_betw (f' \<circ> f) A A''" thus "bij_betw f A A'" using IM proof(auto simp add: bij_betw_def) assume "inj_on (f' \<circ> f) A" thus "inj_on f A" using inj_on_imageI2 by blast next fix a' assume **: "a' \<in> A'" hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using * unfolding bij_betw_def by force hence "f a \<in> A'" using IM by auto hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto thus "a' \<in> f ` A" using 1 by auto qedqedlemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"proof - have i: "inj_on f A" and s: "f ` A = B" using assms by(auto simp:bij_betw_def) let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)" { fix a b assume P: "?P b a" hence ex1: "\<exists>a. ?P b a" using s by blast hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i]) hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp } note g = this have "inj_on ?g B" proof(rule inj_onI) fix x y assume "x:B" "y:B" "?g x = ?g y" from s `x:B` obtain a1 where a1: "?P x a1" by blast from s `y:B` obtain a2 where a2: "?P y a2" by blast from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp qed moreover have "?g ` B = A" proof(auto simp: image_def) fix b assume "b:B" with s obtain a where P: "?P b a" by blast thus "?g b \<in> A" using g[OF P] by auto next fix a assume "a:A" then obtain b where P: "?P b a" using s by blast then have "b:B" using s by blast with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast qed ultimately show ?thesis by(auto simp:bij_betw_def)qedlemma bij_betw_cong: "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"unfolding bij_betw_def inj_on_def by forcelemma bij_betw_id[intro, simp]: "bij_betw id A A"unfolding bij_betw_def id_def by autolemma bij_betw_id_iff: "bij_betw id A B \<longleftrightarrow> A = B"by(auto simp add: bij_betw_def)lemma bij_betw_combine: assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}" shows "bij_betw f (A \<union> C) (B \<union> D)" using assms unfolding bij_betw_def inj_on_Un image_Un by autolemma bij_betw_subset: assumes BIJ: "bij_betw f A A'" and SUB: "B \<le> A" and IM: "f ` B = B'" shows "bij_betw f B B'"using assmsby(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)lemma bij_pointE: assumes "bij f" obtains x where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x"proof - from assms have "inj f" by (rule bij_is_inj) moreover from assms have "surj f" by (rule bij_is_surj) then have "y \<in> range f" by simp ultimately have "\<exists>!x. y = f x" by (simp add: range_ex1_eq) with that show thesis by blastqedlemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"by simplemma surj_vimage_empty: assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}" using surj_image_vimage_eq[OF `surj f`, of A] by (intro iffI) fastforce+lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"by (simp add: inj_on_def, blast)lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"by (blast intro: sym)lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"by (unfold inj_on_def, blast)lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"apply (unfold bij_def)apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)donelemma inj_on_image_eq_iff: "\<lbrakk> inj_on f C; A \<subseteq> C; B \<subseteq> C \<rbrakk> \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"by(fastforce simp add: inj_on_def)lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"by(erule inj_on_image_eq_iff) simp_alllemma inj_on_image_Int: "[| inj_on f C; A<=C; B<=C |] ==> f`(A Int B) = f`A Int f`B"apply (simp add: inj_on_def, blast)donelemma inj_on_image_set_diff: "[| inj_on f C; A<=C; B<=C |] ==> f`(A-B) = f`A - f`B"apply (simp add: inj_on_def, blast)donelemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"by (simp add: inj_on_def, blast)lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"by (simp add: inj_on_def, blast)lemma inj_on_image_mem_iff: "\<lbrakk>inj_on f B; a \<in> B; A \<subseteq> B\<rbrakk> \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A" by (auto simp: inj_on_def)lemma inj_image_mem_iff: "inj f \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A" by (blast dest: injD)lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)" by (blast dest: injD)lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)" by (blast dest: injD)lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"by autolemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"by (auto simp add: inj_on_def)lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"apply (simp add: bij_def)apply (rule equalityI)apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)donelemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}" -- {* The inverse image of a singleton under an injective function is included in a singleton. *} apply (auto simp add: inj_on_def) apply (blast intro: the_equality [symmetric]) donelemma inj_on_vimage_singleton: "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}" by (auto simp add: inj_on_def intro: the_equality [symmetric])lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A" by (auto intro!: inj_onI)lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A" by (auto intro!: inj_onI dest: strict_mono_eq)lemma bij_betw_byWitness:assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A"shows "bij_betw f A A'"using assmsproof(unfold bij_betw_def inj_on_def, safe) fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b" have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp with ** show "a = b" by simpnext fix a' assume *: "a' \<in> A'" hence "f' a' \<in> A" using IM2 by blast moreover have "a' = f(f' a')" using * RIGHT by simp ultimately show "a' \<in> f ` A" by blastqedcorollary notIn_Un_bij_betw:assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and BIJ: "bij_betw f A A'"shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"proof- have "bij_betw f {b} {f b}" unfolding bij_betw_def inj_on_def by simp with assms show ?thesis using bij_betw_combine[of f A A' "{b}" "{f b}"] by blastqedlemma notIn_Un_bij_betw3:assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'"shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"proof assume "bij_betw f A A'" thus "bij_betw f (A \<union> {b}) (A' \<union> {f b})" using assms notIn_Un_bij_betw[of b A f A'] by blastnext assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})" have "f ` A = A'" proof(auto) fix a assume **: "a \<in> A" hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast moreover {assume "f a = f b" hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast with NIN ** have False by blast } ultimately show "f a \<in> A'" by blast next fix a' assume **: "a' \<in> A'" hence "a' \<in> f`(A \<union> {b})" using * by (auto simp add: bij_betw_def) then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast moreover {assume "a = b" with 1 ** NIN' have False by blast } ultimately have "a \<in> A" by blast with 1 show "a' \<in> f ` A" by blast qed thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blastqedsubsection{*Function Updating*}definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where "fun_upd f a b == % x. if x=a then b else f x"nonterminal updbinds and updbindsyntax "_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)") "" :: "updbind => updbinds" ("_") "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _") "_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000, 0] 900)translations "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs" "f(x:=y)" == "CONST fun_upd f x y"(* Hint: to define the sum of two functions (or maps), use case_sum. A nice infix syntax could be defined bynotation case_sum (infixr "'(+')"80)*)lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"apply (simp add: fun_upd_def, safe)apply (erule subst)apply (rule_tac [2] ext, auto)donelemma fun_upd_idem: "f x = y ==> f(x:=y) = f" by (simp only: fun_upd_idem_iff)lemma fun_upd_triv [iff]: "f(x := f x) = f" by (simp only: fun_upd_idem)lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"by (simp add: fun_upd_def)(* fun_upd_apply supersedes these two, but they are useful if fun_upd_apply is intentionally removed from the simpset *)lemma fun_upd_same: "(f(x:=y)) x = y"by simplemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"by simplemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"by (simp add: fun_eq_iff)lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"by (rule ext, auto)lemma inj_on_fun_updI: "inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A" by (fastforce simp: inj_on_def)lemma fun_upd_image: "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"by autolemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)" by autosubsection {* @{text override_on} *}definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"lemma override_on_emptyset[simp]: "override_on f g {} = f"by(simp add:override_on_def)lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"by(simp add:override_on_def)lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"by(simp add:override_on_def)subsection {* @{text swap} *}definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"where "swap a b f = f (a := f b, b:= f a)"lemma swap_apply [simp]: "swap a b f a = f b" "swap a b f b = f a" "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> swap a b f c = f c" by (simp_all add: swap_def)lemma swap_self [simp]: "swap a a f = f" by (simp add: swap_def)lemma swap_commute: "swap a b f = swap b a f" by (simp add: fun_upd_def swap_def fun_eq_iff)lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f" by (rule ext, simp add: fun_upd_def swap_def)lemma swap_comp_involutory [simp]: "swap a b \<circ> swap a b = id" by (rule ext) simplemma swap_triple: assumes "a \<noteq> c" and "b \<noteq> c" shows "swap a b (swap b c (swap a b f)) = swap a c f" using assms by (simp add: fun_eq_iff swap_def)lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)" by (rule ext, simp add: fun_upd_def swap_def)lemma swap_image_eq [simp]: assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"proof - have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A" using assms by (auto simp: image_iff swap_def) then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" . with subset[of f] show ?thesis by autoqedlemma inj_on_imp_inj_on_swap: "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A" by (simp add: inj_on_def swap_def, blast)lemma inj_on_swap_iff [simp]: assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"proof assume "inj_on (swap a b f) A" with A have "inj_on (swap a b (swap a b f)) A" by (iprover intro: inj_on_imp_inj_on_swap) thus "inj_on f A" by simpnext assume "inj_on f A" with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)qedlemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)" by simplemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f" by simplemma bij_betw_swap_iff [simp]: "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B" by (auto simp: bij_betw_def)lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f" by simphide_const (open) swapsubsection {* Inversion of injective functions *}definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where "the_inv_into A f == %x. THE y. y : A & f y = x"lemma the_inv_into_f_f: "[| inj_on f A; x : A |] ==> the_inv_into A f (f x) = x"apply (simp add: the_inv_into_def inj_on_def)apply blastdonelemma f_the_inv_into_f: "inj_on f A ==> y : f`A ==> f (the_inv_into A f y) = y"apply (simp add: the_inv_into_def)apply (rule the1I2) apply(blast dest: inj_onD)apply blastdonelemma the_inv_into_into: "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"apply (simp add: the_inv_into_def)apply (rule the1I2) apply(blast dest: inj_onD)apply blastdonelemma the_inv_into_onto[simp]: "inj_on f A ==> the_inv_into A f ` (f ` A) = A"by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])lemma the_inv_into_f_eq: "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x" apply (erule subst) apply (erule the_inv_into_f_f, assumption) donelemma the_inv_into_comp: "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==> the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"apply (rule the_inv_into_f_eq) apply (fast intro: comp_inj_on) apply (simp add: f_the_inv_into_f the_inv_into_into)apply (simp add: the_inv_into_into)donelemma inj_on_the_inv_into: "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"by (auto intro: inj_onI simp: the_inv_into_f_f)lemma bij_betw_the_inv_into: "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where "the_inv f \<equiv> the_inv_into UNIV f"lemma the_inv_f_f: assumes "inj f" shows "the_inv f (f x) = x" using assms UNIV_I by (rule the_inv_into_f_f)subsection {* Cantor's Paradox *}lemma Cantors_paradox: "\<not>(\<exists>f. f ` A = Pow A)"proof clarify fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast let ?X = "{a \<in> A. a \<notin> f a}" have "?X \<in> Pow A" unfolding Pow_def by auto with * obtain x where "x \<in> A \<and> f x = ?X" by blast thus False by bestqedsubsection {* Setup *} subsubsection {* Proof tools *}text {* simplifies terms of the form f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>let fun gen_fun_upd NONE T _ _ = NONE | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y) fun dest_fun_T1 (Type (_, T :: Ts)) = T fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) = let fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) = if v aconv x then SOME g else gen_fun_upd (find g) T v w | find t = NONE in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end val ss = simpset_of @{context} fun proc ctxt ct = let val t = Thm.term_of ct in case find_double t of (T, NONE) => NONE | (T, SOME rhs) => SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs)) (fn _ => resolve_tac ctxt [eq_reflection] 1 THEN resolve_tac ctxt @{thms ext} 1 THEN simp_tac (put_simpset ss ctxt) 1)) endin proc end*}subsubsection {* Functorial structure of types *}ML_file "Tools/functor.ML"functor map_fun: map_fun by (simp_all add: fun_eq_iff)functor vimage by (simp_all add: fun_eq_iff vimage_comp)text {* Legacy theorem names *}lemmas o_def = comp_deflemmas o_apply = comp_applylemmas o_assoc = comp_assoc [symmetric]lemmas id_o = id_complemmas o_id = comp_idlemmas o_eq_dest = comp_eq_destlemmas o_eq_elim = comp_eq_elimlemmas o_eq_dest_lhs = comp_eq_dest_lhslemmas o_eq_id_dest = comp_eq_id_destend