src/HOL/Fun.thy
author wenzelm
Thu Mar 14 16:55:06 2019 +0100 (5 weeks ago)
changeset 69913 ca515cf61651
parent 69768 7e4966eaf781
permissions -rw-r--r--
more specific keyword kinds;
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(*  Title:      HOL/Fun.thy
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    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
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    Author:     Andrei Popescu, TU Muenchen
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    Copyright   1994, 2012
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*)
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section \<open>Notions about functions\<close>
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theory Fun
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  imports Set
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  keywords "functor" :: thy_goal_defn
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begin
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lemma apply_inverse: "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
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  by auto
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text \<open>Uniqueness, so NOT the axiom of choice.\<close>
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lemma uniq_choice: "\<forall>x. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)"
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  by (force intro: theI')
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lemma b_uniq_choice: "\<forall>x\<in>S. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x\<in>S. Q x (f x)"
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  by (force intro: theI')
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subsection \<open>The Identity Function \<open>id\<close>\<close>
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definition id :: "'a \<Rightarrow> 'a"
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  where "id = (\<lambda>x. x)"
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lemma id_apply [simp]: "id x = x"
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  by (simp add: id_def)
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lemma image_id [simp]: "image id = id"
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  by (simp add: id_def fun_eq_iff)
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lemma vimage_id [simp]: "vimage id = id"
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  by (simp add: id_def fun_eq_iff)
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lemma eq_id_iff: "(\<forall>x. f x = x) \<longleftrightarrow> f = id"
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  by auto
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code_printing
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  constant id \<rightharpoonup> (Haskell) "id"
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subsection \<open>The Composition Operator \<open>f \<circ> g\<close>\<close>
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definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c"  (infixl "\<circ>" 55)
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  where "f \<circ> g = (\<lambda>x. f (g x))"
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notation (ASCII)
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  comp  (infixl "o" 55)
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lemma comp_apply [simp]: "(f \<circ> g) x = f (g x)"
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  by (simp add: comp_def)
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lemma comp_assoc: "(f \<circ> g) \<circ> h = f \<circ> (g \<circ> h)"
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  by (simp add: fun_eq_iff)
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lemma id_comp [simp]: "id \<circ> g = g"
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  by (simp add: fun_eq_iff)
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lemma comp_id [simp]: "f \<circ> id = f"
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  by (simp add: fun_eq_iff)
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lemma comp_eq_dest: "a \<circ> b = c \<circ> d \<Longrightarrow> a (b v) = c (d v)"
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  by (simp add: fun_eq_iff)
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lemma comp_eq_elim: "a \<circ> b = c \<circ> d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
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  by (simp add: fun_eq_iff)
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lemma comp_eq_dest_lhs: "a \<circ> b = c \<Longrightarrow> a (b v) = c v"
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  by clarsimp
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lemma comp_eq_id_dest: "a \<circ> b = id \<circ> c \<Longrightarrow> a (b v) = c v"
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  by clarsimp
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lemma image_comp: "f ` (g ` r) = (f \<circ> g) ` r"
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  by auto
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lemma vimage_comp: "f -` (g -` x) = (g \<circ> f) -` x"
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  by auto
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lemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h \<circ> f) ` A = (h \<circ> g) ` B"
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  by (auto simp: comp_def elim!: equalityE)
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lemma image_bind: "f ` (Set.bind A g) = Set.bind A ((`) f \<circ> g)"
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  by (auto simp add: Set.bind_def)
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lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g \<circ> f)"
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  by (auto simp add: Set.bind_def)
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lemma (in group_add) minus_comp_minus [simp]: "uminus \<circ> uminus = id"
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  by (simp add: fun_eq_iff)
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lemma (in boolean_algebra) minus_comp_minus [simp]: "uminus \<circ> uminus = id"
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  by (simp add: fun_eq_iff)
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code_printing
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  constant comp \<rightharpoonup> (SML) infixl 5 "o" and (Haskell) infixr 9 "."
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subsection \<open>The Forward Composition Operator \<open>fcomp\<close>\<close>
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definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c"  (infixl "\<circ>>" 60)
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  where "f \<circ>> g = (\<lambda>x. g (f x))"
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lemma fcomp_apply [simp]:  "(f \<circ>> g) x = g (f x)"
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  by (simp add: fcomp_def)
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lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)"
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  by (simp add: fcomp_def)
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lemma id_fcomp [simp]: "id \<circ>> g = g"
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  by (simp add: fcomp_def)
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lemma fcomp_id [simp]: "f \<circ>> id = f"
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  by (simp add: fcomp_def)
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lemma fcomp_comp: "fcomp f g = comp g f"
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  by (simp add: ext)
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code_printing
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  constant fcomp \<rightharpoonup> (Eval) infixl 1 "#>"
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no_notation fcomp (infixl "\<circ>>" 60)
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subsection \<open>Mapping functions\<close>
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definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd"
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  where "map_fun f g h = g \<circ> h \<circ> f"
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lemma map_fun_apply [simp]: "map_fun f g h x = g (h (f x))"
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  by (simp add: map_fun_def)
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subsection \<open>Injectivity and Bijectivity\<close>
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definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool"  \<comment> \<open>injective\<close>
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  where "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
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definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"  \<comment> \<open>bijective\<close>
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  where "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
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text \<open>
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  A common special case: functions injective, surjective or bijective over
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  the entire domain type.
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\<close>
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abbreviation inj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"
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  where "inj f \<equiv> inj_on f UNIV"
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abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"
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  where "surj f \<equiv> range f = UNIV"
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translations \<comment> \<open>The negated case:\<close>
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  "\<not> CONST surj f" \<leftharpoondown> "CONST range f \<noteq> CONST UNIV"
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abbreviation bij :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"
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  where "bij f \<equiv> bij_betw f UNIV UNIV"
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lemma inj_def: "inj f \<longleftrightarrow> (\<forall>x y. f x = f y \<longrightarrow> x = y)"
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  unfolding inj_on_def by blast
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lemma injI: "(\<And>x y. f x = f y \<Longrightarrow> x = y) \<Longrightarrow> inj f"
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  unfolding inj_def by blast
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theorem range_ex1_eq: "inj f \<Longrightarrow> b \<in> range f \<longleftrightarrow> (\<exists>!x. b = f x)"
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  unfolding inj_def by blast
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lemma injD: "inj f \<Longrightarrow> f x = f y \<Longrightarrow> x = y"
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  by (simp add: inj_def)
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lemma inj_on_eq_iff: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<longleftrightarrow> x = y"
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  by (auto simp: inj_on_def)
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lemma inj_on_cong: "(\<And>a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A \<longleftrightarrow> inj_on g A"
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  by (auto simp: inj_on_def)
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lemma inj_on_strict_subset: "inj_on f B \<Longrightarrow> A \<subset> B \<Longrightarrow> f ` A \<subset> f ` B"
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  unfolding inj_on_def by blast
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lemma inj_compose: "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
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  by (simp add: inj_def)
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lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"
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  by (simp add: inj_def fun_eq_iff)
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lemma inj_eq: "inj f \<Longrightarrow> f x = f y \<longleftrightarrow> x = y"
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  by (simp add: inj_on_eq_iff)
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lemma inj_on_id[simp]: "inj_on id A"
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  by (simp add: inj_on_def)
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lemma inj_on_id2[simp]: "inj_on (\<lambda>x. x) A"
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  by (simp add: inj_on_def)
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lemma inj_on_Int: "inj_on f A \<or> inj_on f B \<Longrightarrow> inj_on f (A \<inter> B)"
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  unfolding inj_on_def by blast
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lemma surj_id: "surj id"
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  by simp
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lemma bij_id[simp]: "bij id"
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  by (simp add: bij_betw_def)
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lemma bij_uminus: "bij (uminus :: 'a \<Rightarrow> 'a::ab_group_add)"
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  unfolding bij_betw_def inj_on_def
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  by (force intro: minus_minus [symmetric])
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lemma inj_onI [intro?]: "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<Longrightarrow> x = y) \<Longrightarrow> inj_on f A"
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  by (simp add: inj_on_def)
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lemma inj_on_inverseI: "(\<And>x. x \<in> A \<Longrightarrow> g (f x) = x) \<Longrightarrow> inj_on f A"
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  by (auto dest: arg_cong [of concl: g] simp add: inj_on_def)
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lemma inj_onD: "inj_on f A \<Longrightarrow> f x = f y \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y"
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  unfolding inj_on_def by blast
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lemma inj_on_subset:
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  assumes "inj_on f A"
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    and "B \<subseteq> A"
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  shows "inj_on f B"
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proof (rule inj_onI)
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  fix a b
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  assume "a \<in> B" and "b \<in> B"
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  with assms have "a \<in> A" and "b \<in> A"
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    by auto
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  moreover assume "f a = f b"
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  ultimately show "a = b"
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    using assms by (auto dest: inj_onD)
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qed
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lemma comp_inj_on: "inj_on f A \<Longrightarrow> inj_on g (f ` A) \<Longrightarrow> inj_on (g \<circ> f) A"
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  by (simp add: comp_def inj_on_def)
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lemma inj_on_imageI: "inj_on (g \<circ> f) A \<Longrightarrow> inj_on g (f ` A)"
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  by (auto simp add: inj_on_def)
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lemma inj_on_image_iff:
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  "\<forall>x\<in>A. \<forall>y\<in>A. g (f x) = g (f y) \<longleftrightarrow> g x = g y \<Longrightarrow> inj_on f A \<Longrightarrow> inj_on g (f ` A) \<longleftrightarrow> inj_on g A"
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  unfolding inj_on_def by blast
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lemma inj_on_contraD: "inj_on f A \<Longrightarrow> x \<noteq> y \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x \<noteq> f y"
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  unfolding inj_on_def by blast
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lemma inj_singleton [simp]: "inj_on (\<lambda>x. {x}) A"
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  by (simp add: inj_on_def)
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lemma inj_on_empty[iff]: "inj_on f {}"
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  by (simp add: inj_on_def)
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lemma subset_inj_on: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> inj_on f A"
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  unfolding inj_on_def by blast
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lemma inj_on_Un: "inj_on f (A \<union> B) \<longleftrightarrow> inj_on f A \<and> inj_on f B \<and> f ` (A - B) \<inter> f ` (B - A) = {}"
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  unfolding inj_on_def by (blast intro: sym)
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lemma inj_on_insert [iff]: "inj_on f (insert a A) \<longleftrightarrow> inj_on f A \<and> f a \<notin> f ` (A - {a})"
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  unfolding inj_on_def by (blast intro: sym)
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lemma inj_on_diff: "inj_on f A \<Longrightarrow> inj_on f (A - B)"
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  unfolding inj_on_def by blast
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lemma comp_inj_on_iff: "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' \<circ> f) A"
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  by (auto simp: comp_inj_on inj_on_def)
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lemma inj_on_imageI2: "inj_on (f' \<circ> f) A \<Longrightarrow> inj_on f A"
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  by (auto simp: comp_inj_on inj_on_def)
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lemma inj_img_insertE:
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  assumes "inj_on f A"
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  assumes "x \<notin> B"
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    and "insert x B = f ` A"
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  obtains x' A' where "x' \<notin> A'" and "A = insert x' A'" and "x = f x'" and "B = f ` A'"
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proof -
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  from assms have "x \<in> f ` A" by auto
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  then obtain x' where *: "x' \<in> A" "x = f x'" by auto
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  then have A: "A = insert x' (A - {x'})" by auto
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  with assms * have B: "B = f ` (A - {x'})" by (auto dest: inj_on_contraD)
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  have "x' \<notin> A - {x'}" by simp
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  from this A \<open>x = f x'\<close> B show ?thesis ..
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qed
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lemma linorder_injI:
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  assumes "\<And>x y::'a::linorder. x < y \<Longrightarrow> f x \<noteq> f y"
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  shows "inj f"
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  \<comment> \<open>Courtesy of Stephan Merz\<close>
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proof (rule inj_onI)
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  show "x = y" if "f x = f y" for x y
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   by (rule linorder_cases) (auto dest: assms simp: that)
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qed
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lemma inj_on_image_Pow: "inj_on f A \<Longrightarrow>inj_on (image f) (Pow A)"
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  unfolding Pow_def inj_on_def by blast
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lemma bij_betw_image_Pow: "bij_betw f A B \<Longrightarrow> bij_betw (image f) (Pow A) (Pow B)"
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  by (auto simp add: bij_betw_def inj_on_image_Pow image_Pow_surj)
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lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
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  by auto
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   305
lemma surjI:
wenzelm@64965
   306
  assumes "\<And>x. g (f x) = x"
wenzelm@63322
   307
  shows "surj g"
wenzelm@64965
   308
  using assms [symmetric] by auto
paulson@13585
   309
hoelzl@39076
   310
lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
hoelzl@39076
   311
  by (simp add: surj_def)
paulson@13585
   312
hoelzl@39076
   313
lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
wenzelm@63575
   314
  by (simp add: surj_def) blast
paulson@13585
   315
wenzelm@63322
   316
lemma comp_surj: "surj f \<Longrightarrow> surj g \<Longrightarrow> surj (g \<circ> f)"
haftmann@69768
   317
  using image_comp [of g f UNIV] by simp
paulson@13585
   318
wenzelm@63322
   319
lemma bij_betw_imageI: "inj_on f A \<Longrightarrow> f ` A = B \<Longrightarrow> bij_betw f A B"
wenzelm@63322
   320
  unfolding bij_betw_def by clarify
ballarin@57282
   321
ballarin@57282
   322
lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> f ` A = B"
ballarin@57282
   323
  unfolding bij_betw_def by clarify
ballarin@57282
   324
hoelzl@39074
   325
lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
hoelzl@40702
   326
  unfolding bij_betw_def by auto
hoelzl@39074
   327
wenzelm@63322
   328
lemma bij_betw_empty1: "bij_betw f {} A \<Longrightarrow> A = {}"
wenzelm@63322
   329
  unfolding bij_betw_def by blast
hoelzl@40703
   330
wenzelm@63322
   331
lemma bij_betw_empty2: "bij_betw f A {} \<Longrightarrow> A = {}"
wenzelm@63322
   332
  unfolding bij_betw_def by blast
hoelzl@40703
   333
wenzelm@63322
   334
lemma inj_on_imp_bij_betw: "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
wenzelm@63322
   335
  unfolding bij_betw_def by simp
hoelzl@40703
   336
hoelzl@39076
   337
lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
wenzelm@64965
   338
  by (rule bij_betw_def)
hoelzl@39074
   339
wenzelm@63322
   340
lemma bijI: "inj f \<Longrightarrow> surj f \<Longrightarrow> bij f"
wenzelm@64965
   341
  by (rule bij_betw_imageI)
paulson@13585
   342
wenzelm@63322
   343
lemma bij_is_inj: "bij f \<Longrightarrow> inj f"
wenzelm@63322
   344
  by (simp add: bij_def)
paulson@13585
   345
wenzelm@63322
   346
lemma bij_is_surj: "bij f \<Longrightarrow> surj f"
wenzelm@63322
   347
  by (simp add: bij_def)
paulson@13585
   348
nipkow@26105
   349
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
wenzelm@63322
   350
  by (simp add: bij_betw_def)
nipkow@26105
   351
wenzelm@63322
   352
lemma bij_betw_trans: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g \<circ> f) A C"
wenzelm@63322
   353
  by (auto simp add:bij_betw_def comp_inj_on)
nipkow@31438
   354
wenzelm@63322
   355
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g \<circ> f)"
hoelzl@40702
   356
  by (rule bij_betw_trans)
hoelzl@40702
   357
wenzelm@63322
   358
lemma bij_betw_comp_iff: "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' \<circ> f) A A''"
wenzelm@63322
   359
  by (auto simp add: bij_betw_def inj_on_def)
hoelzl@40703
   360
hoelzl@40703
   361
lemma bij_betw_comp_iff2:
wenzelm@63322
   362
  assumes bij: "bij_betw f' A' A''"
wenzelm@63322
   363
    and img: "f ` A \<le> A'"
wenzelm@63322
   364
  shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' \<circ> f) A A''"
wenzelm@63322
   365
  using assms
wenzelm@63322
   366
proof (auto simp add: bij_betw_comp_iff)
hoelzl@40703
   367
  assume *: "bij_betw (f' \<circ> f) A A''"
wenzelm@63322
   368
  then show "bij_betw f A A'"
wenzelm@63322
   369
    using img
wenzelm@63322
   370
  proof (auto simp add: bij_betw_def)
hoelzl@40703
   371
    assume "inj_on (f' \<circ> f) A"
wenzelm@63575
   372
    then show "inj_on f A"
wenzelm@63575
   373
      using inj_on_imageI2 by blast
hoelzl@40703
   374
  next
wenzelm@63322
   375
    fix a'
wenzelm@63322
   376
    assume **: "a' \<in> A'"
wenzelm@63575
   377
    with bij have "f' a' \<in> A''"
wenzelm@63575
   378
      unfolding bij_betw_def by auto
wenzelm@63575
   379
    with * obtain a where 1: "a \<in> A \<and> f' (f a) = f' a'"
wenzelm@63575
   380
      unfolding bij_betw_def by force
wenzelm@63575
   381
    with img have "f a \<in> A'" by auto
wenzelm@63575
   382
    with bij ** 1 have "f a = a'"
wenzelm@63575
   383
      unfolding bij_betw_def inj_on_def by auto
wenzelm@63575
   384
    with 1 show "a' \<in> f ` A" by auto
hoelzl@40703
   385
  qed
hoelzl@40703
   386
qed
hoelzl@40703
   387
wenzelm@63322
   388
lemma bij_betw_inv:
wenzelm@63322
   389
  assumes "bij_betw f A B"
wenzelm@63322
   390
  shows "\<exists>g. bij_betw g B A"
nipkow@26105
   391
proof -
nipkow@26105
   392
  have i: "inj_on f A" and s: "f ` A = B"
wenzelm@63322
   393
    using assms by (auto simp: bij_betw_def)
wenzelm@63322
   394
  let ?P = "\<lambda>b a. a \<in> A \<and> f a = b"
wenzelm@63322
   395
  let ?g = "\<lambda>b. The (?P b)"
wenzelm@63322
   396
  have g: "?g b = a" if P: "?P b a" for a b
wenzelm@63322
   397
  proof -
wenzelm@63575
   398
    from that s have ex1: "\<exists>a. ?P b a" by blast
wenzelm@63322
   399
    then have uex1: "\<exists>!a. ?P b a" by (blast dest:inj_onD[OF i])
wenzelm@63575
   400
    then show ?thesis
wenzelm@63575
   401
      using the1_equality[OF uex1, OF P] P by simp
wenzelm@63322
   402
  qed
nipkow@26105
   403
  have "inj_on ?g B"
wenzelm@63322
   404
  proof (rule inj_onI)
wenzelm@63322
   405
    fix x y
wenzelm@63322
   406
    assume "x \<in> B" "y \<in> B" "?g x = ?g y"
wenzelm@63322
   407
    from s \<open>x \<in> B\<close> obtain a1 where a1: "?P x a1" by blast
wenzelm@63322
   408
    from s \<open>y \<in> B\<close> obtain a2 where a2: "?P y a2" by blast
wenzelm@63322
   409
    from g [OF a1] a1 g [OF a2] a2 \<open>?g x = ?g y\<close> show "x = y" by simp
nipkow@26105
   410
  qed
nipkow@26105
   411
  moreover have "?g ` B = A"
wenzelm@63322
   412
  proof (auto simp: image_def)
wenzelm@63322
   413
    fix b
wenzelm@63322
   414
    assume "b \<in> B"
haftmann@56077
   415
    with s obtain a where P: "?P b a" by blast
wenzelm@63575
   416
    with g[OF P] show "?g b \<in> A" by auto
nipkow@26105
   417
  next
wenzelm@63322
   418
    fix a
wenzelm@63322
   419
    assume "a \<in> A"
wenzelm@63575
   420
    with s obtain b where P: "?P b a" by blast
wenzelm@63575
   421
    with s have "b \<in> B" by blast
nipkow@26105
   422
    with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
nipkow@26105
   423
  qed
wenzelm@63575
   424
  ultimately show ?thesis
wenzelm@63575
   425
    by (auto simp: bij_betw_def)
nipkow@26105
   426
qed
nipkow@26105
   427
wenzelm@63588
   428
lemma bij_betw_cong: "(\<And>a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
wenzelm@63591
   429
  unfolding bij_betw_def inj_on_def by safe force+  (* somewhat slow *)
hoelzl@40703
   430
wenzelm@63322
   431
lemma bij_betw_id[intro, simp]: "bij_betw id A A"
wenzelm@63322
   432
  unfolding bij_betw_def id_def by auto
hoelzl@40703
   433
wenzelm@63322
   434
lemma bij_betw_id_iff: "bij_betw id A B \<longleftrightarrow> A = B"
wenzelm@63322
   435
  by (auto simp add: bij_betw_def)
hoelzl@40703
   436
hoelzl@39075
   437
lemma bij_betw_combine:
wenzelm@63400
   438
  "bij_betw f A B \<Longrightarrow> bij_betw f C D \<Longrightarrow> B \<inter> D = {} \<Longrightarrow> bij_betw f (A \<union> C) (B \<union> D)"
wenzelm@63400
   439
  unfolding bij_betw_def inj_on_Un image_Un by auto
hoelzl@39075
   440
wenzelm@64966
   441
lemma bij_betw_subset: "bij_betw f A A' \<Longrightarrow> B \<subseteq> A \<Longrightarrow> f ` B = B' \<Longrightarrow> bij_betw f B B'"
wenzelm@63322
   442
  by (auto simp add: bij_betw_def inj_on_def)
hoelzl@40703
   443
haftmann@58195
   444
lemma bij_pointE:
haftmann@58195
   445
  assumes "bij f"
haftmann@58195
   446
  obtains x where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x"
haftmann@58195
   447
proof -
haftmann@58195
   448
  from assms have "inj f" by (rule bij_is_inj)
haftmann@58195
   449
  moreover from assms have "surj f" by (rule bij_is_surj)
haftmann@58195
   450
  then have "y \<in> range f" by simp
haftmann@58195
   451
  ultimately have "\<exists>!x. y = f x" by (simp add: range_ex1_eq)
haftmann@58195
   452
  with that show thesis by blast
haftmann@58195
   453
qed
haftmann@58195
   454
wenzelm@63322
   455
lemma surj_image_vimage_eq: "surj f \<Longrightarrow> f ` (f -` A) = A"
wenzelm@63322
   456
  by simp
paulson@13585
   457
hoelzl@42903
   458
lemma surj_vimage_empty:
wenzelm@63322
   459
  assumes "surj f"
wenzelm@63322
   460
  shows "f -` A = {} \<longleftrightarrow> A = {}"
wenzelm@63322
   461
  using surj_image_vimage_eq [OF \<open>surj f\<close>, of A]
nipkow@44890
   462
  by (intro iffI) fastforce+
hoelzl@42903
   463
wenzelm@63322
   464
lemma inj_vimage_image_eq: "inj f \<Longrightarrow> f -` (f ` A) = A"
wenzelm@64966
   465
  unfolding inj_def by blast
paulson@13585
   466
wenzelm@63322
   467
lemma vimage_subsetD: "surj f \<Longrightarrow> f -` B \<subseteq> A \<Longrightarrow> B \<subseteq> f ` A"
wenzelm@63322
   468
  by (blast intro: sym)
paulson@13585
   469
wenzelm@63322
   470
lemma vimage_subsetI: "inj f \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> f -` B \<subseteq> A"
wenzelm@64966
   471
  unfolding inj_def by blast
paulson@13585
   472
wenzelm@63322
   473
lemma vimage_subset_eq: "bij f \<Longrightarrow> f -` B \<subseteq> A \<longleftrightarrow> B \<subseteq> f ` A"
wenzelm@63322
   474
  unfolding bij_def by (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
paulson@13585
   475
wenzelm@63322
   476
lemma inj_on_image_eq_iff: "inj_on f C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
wenzelm@64965
   477
  by (fastforce simp: inj_on_def)
Andreas@53927
   478
nipkow@31438
   479
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
wenzelm@63322
   480
  by (erule inj_on_image_eq_iff) simp_all
nipkow@31438
   481
wenzelm@63322
   482
lemma inj_on_image_Int: "inj_on f C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` (A \<inter> B) = f ` A \<inter> f ` B"
wenzelm@63322
   483
  unfolding inj_on_def by blast
wenzelm@63322
   484
wenzelm@63322
   485
lemma inj_on_image_set_diff: "inj_on f C \<Longrightarrow> A - B \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` (A - B) = f ` A - f ` B"
wenzelm@63322
   486
  unfolding inj_on_def by blast
paulson@13585
   487
wenzelm@63322
   488
lemma image_Int: "inj f \<Longrightarrow> f ` (A \<inter> B) = f ` A \<inter> f ` B"
wenzelm@64966
   489
  unfolding inj_def by blast
paulson@13585
   490
wenzelm@63322
   491
lemma image_set_diff: "inj f \<Longrightarrow> f ` (A - B) = f ` A - f ` B"
wenzelm@64966
   492
  unfolding inj_def by blast
paulson@13585
   493
wenzelm@63322
   494
lemma inj_on_image_mem_iff: "inj_on f B \<Longrightarrow> a \<in> B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f ` A \<longleftrightarrow> a \<in> A"
lp15@59504
   495
  by (auto simp: inj_on_def)
lp15@59504
   496
lp15@61520
   497
(*FIXME DELETE*)
wenzelm@63322
   498
lemma inj_on_image_mem_iff_alt: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f ` A \<Longrightarrow> a \<in> B \<Longrightarrow> a \<in> A"
lp15@61520
   499
  by (blast dest: inj_onD)
lp15@61520
   500
wenzelm@63322
   501
lemma inj_image_mem_iff: "inj f \<Longrightarrow> f a \<in> f ` A \<longleftrightarrow> a \<in> A"
lp15@59504
   502
  by (blast dest: injD)
paulson@13585
   503
wenzelm@63322
   504
lemma inj_image_subset_iff: "inj f \<Longrightarrow> f ` A \<subseteq> f ` B \<longleftrightarrow> A \<subseteq> B"
lp15@59504
   505
  by (blast dest: injD)
paulson@13585
   506
wenzelm@63322
   507
lemma inj_image_eq_iff: "inj f \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
lp15@59504
   508
  by (blast dest: injD)
paulson@13585
   509
wenzelm@63322
   510
lemma surj_Compl_image_subset: "surj f \<Longrightarrow> - (f ` A) \<subseteq> f ` (- A)"
wenzelm@63322
   511
  by auto
paulson@5852
   512
wenzelm@63322
   513
lemma inj_image_Compl_subset: "inj f \<Longrightarrow> f ` (- A) \<subseteq> - (f ` A)"
wenzelm@64966
   514
  by (auto simp: inj_def)
wenzelm@63322
   515
wenzelm@63322
   516
lemma bij_image_Compl_eq: "bij f \<Longrightarrow> f ` (- A) = - (f ` A)"
wenzelm@63322
   517
  by (simp add: bij_def inj_image_Compl_subset surj_Compl_image_subset equalityI)
paulson@13585
   518
haftmann@41657
   519
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
wenzelm@63322
   520
  \<comment> \<open>The inverse image of a singleton under an injective function is included in a singleton.\<close>
wenzelm@64966
   521
  by (simp add: inj_def) (blast intro: the_equality [symmetric])
haftmann@41657
   522
wenzelm@63322
   523
lemma inj_on_vimage_singleton: "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
hoelzl@43991
   524
  by (auto simp add: inj_on_def intro: the_equality [symmetric])
hoelzl@43991
   525
hoelzl@35584
   526
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
hoelzl@35580
   527
  by (auto intro!: inj_onI)
paulson@13585
   528
hoelzl@35584
   529
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
hoelzl@35584
   530
  by (auto intro!: inj_onI dest: strict_mono_eq)
hoelzl@35584
   531
blanchet@55019
   532
lemma bij_betw_byWitness:
wenzelm@63322
   533
  assumes left: "\<forall>a \<in> A. f' (f a) = a"
wenzelm@63322
   534
    and right: "\<forall>a' \<in> A'. f (f' a') = a'"
wenzelm@63575
   535
    and "f ` A \<subseteq> A'"
wenzelm@63575
   536
    and img2: "f' ` A' \<subseteq> A"
wenzelm@63322
   537
  shows "bij_betw f A A'"
wenzelm@63322
   538
  using assms
wenzelm@63400
   539
  unfolding bij_betw_def inj_on_def
wenzelm@63400
   540
proof safe
wenzelm@63322
   541
  fix a b
wenzelm@63575
   542
  assume "a \<in> A" "b \<in> A"
wenzelm@63575
   543
  with left have "a = f' (f a) \<and> b = f' (f b)" by simp
wenzelm@63575
   544
  moreover assume "f a = f b"
wenzelm@63575
   545
  ultimately show "a = b" by simp
blanchet@55019
   546
next
blanchet@55019
   547
  fix a' assume *: "a' \<in> A'"
wenzelm@63575
   548
  with img2 have "f' a' \<in> A" by blast
wenzelm@63575
   549
  moreover from * right have "a' = f (f' a')" by simp
blanchet@55019
   550
  ultimately show "a' \<in> f ` A" by blast
blanchet@55019
   551
qed
blanchet@55019
   552
blanchet@55019
   553
corollary notIn_Un_bij_betw:
wenzelm@63322
   554
  assumes "b \<notin> A"
wenzelm@63322
   555
    and "f b \<notin> A'"
wenzelm@63322
   556
    and "bij_betw f A A'"
wenzelm@63322
   557
  shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
wenzelm@63322
   558
proof -
blanchet@55019
   559
  have "bij_betw f {b} {f b}"
wenzelm@63322
   560
    unfolding bij_betw_def inj_on_def by simp
blanchet@55019
   561
  with assms show ?thesis
wenzelm@63322
   562
    using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
blanchet@55019
   563
qed
blanchet@55019
   564
blanchet@55019
   565
lemma notIn_Un_bij_betw3:
wenzelm@63322
   566
  assumes "b \<notin> A"
wenzelm@63322
   567
    and "f b \<notin> A'"
wenzelm@63322
   568
  shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   569
proof
blanchet@55019
   570
  assume "bij_betw f A A'"
wenzelm@63322
   571
  then show "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
wenzelm@63322
   572
    using assms notIn_Un_bij_betw [of b A f A'] by blast
blanchet@55019
   573
next
blanchet@55019
   574
  assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   575
  have "f ` A = A'"
wenzelm@63322
   576
  proof auto
wenzelm@63322
   577
    fix a
wenzelm@63322
   578
    assume **: "a \<in> A"
wenzelm@63322
   579
    then have "f a \<in> A' \<union> {f b}"
wenzelm@63322
   580
      using * unfolding bij_betw_def by blast
blanchet@55019
   581
    moreover
wenzelm@63322
   582
    have False if "f a = f b"
wenzelm@63322
   583
    proof -
wenzelm@63575
   584
      have "a = b"
wenzelm@63575
   585
        using * ** that unfolding bij_betw_def inj_on_def by blast
wenzelm@63322
   586
      with \<open>b \<notin> A\<close> ** show ?thesis by blast
wenzelm@63322
   587
    qed
blanchet@55019
   588
    ultimately show "f a \<in> A'" by blast
blanchet@55019
   589
  next
wenzelm@63322
   590
    fix a'
wenzelm@63322
   591
    assume **: "a' \<in> A'"
wenzelm@63322
   592
    then have "a' \<in> f ` (A \<union> {b})"
wenzelm@63322
   593
      using * by (auto simp add: bij_betw_def)
blanchet@55019
   594
    then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast
blanchet@55019
   595
    moreover
wenzelm@63322
   596
    have False if "a = b" using 1 ** \<open>f b \<notin> A'\<close> that by blast
blanchet@55019
   597
    ultimately have "a \<in> A" by blast
blanchet@55019
   598
    with 1 show "a' \<in> f ` A" by blast
blanchet@55019
   599
  qed
wenzelm@63322
   600
  then show "bij_betw f A A'"
wenzelm@63322
   601
    using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
blanchet@55019
   602
qed
blanchet@55019
   603
haftmann@69502
   604
text \<open>Important examples\<close>
haftmann@69502
   605
haftmann@69502
   606
context cancel_semigroup_add
haftmann@69502
   607
begin
haftmann@69502
   608
haftmann@69661
   609
lemma inj_on_add [simp]:
haftmann@69661
   610
  "inj_on ((+) a) A"
haftmann@69661
   611
  by (rule inj_onI) simp
haftmann@69661
   612
haftmann@69661
   613
lemma inj_add_left:
haftmann@69661
   614
  \<open>inj ((+) a)\<close>
haftmann@69661
   615
  by simp
haftmann@69661
   616
haftmann@69661
   617
lemma inj_on_add' [simp]:
haftmann@69661
   618
  "inj_on (\<lambda>b. b + a) A"
haftmann@69661
   619
  by (rule inj_onI) simp
haftmann@69661
   620
haftmann@69661
   621
lemma bij_betw_add [simp]:
haftmann@69661
   622
  "bij_betw ((+) a) A B \<longleftrightarrow> (+) a ` A = B"
haftmann@69661
   623
  by (simp add: bij_betw_def)
haftmann@69502
   624
haftmann@69502
   625
end
haftmann@69502
   626
haftmann@69502
   627
context ab_group_add
haftmann@69502
   628
begin
haftmann@69502
   629
haftmann@69661
   630
lemma surj_plus [simp]:
haftmann@69661
   631
  "surj ((+) a)"
haftmann@69768
   632
  by (auto intro!: range_eqI [of b "(+) a" "b - a" for b]) (simp add: algebra_simps)
haftmann@69661
   633
haftmann@69661
   634
lemma inj_diff_right [simp]:
haftmann@69661
   635
  \<open>inj (\<lambda>b. b - a)\<close>
haftmann@69502
   636
proof -
haftmann@69502
   637
  have \<open>inj ((+) (- a))\<close>
haftmann@69502
   638
    by (fact inj_add_left)
haftmann@69502
   639
  also have \<open>(+) (- a) = (\<lambda>b. b - a)\<close>
haftmann@69502
   640
    by (simp add: fun_eq_iff)
haftmann@69502
   641
  finally show ?thesis .
haftmann@69502
   642
qed
haftmann@69502
   643
haftmann@69661
   644
lemma surj_diff_right [simp]:
haftmann@69661
   645
  "surj (\<lambda>x. x - a)"
haftmann@69661
   646
  using surj_plus [of "- a"] by (simp cong: image_cong_simp)
haftmann@69661
   647
haftmann@69661
   648
lemma translation_Compl:
haftmann@69661
   649
  "(+) a ` (- t) = - ((+) a ` t)"
haftmann@69661
   650
proof (rule set_eqI)
haftmann@69661
   651
  fix b
haftmann@69661
   652
  show "b \<in> (+) a ` (- t) \<longleftrightarrow> b \<in> - (+) a ` t"
haftmann@69661
   653
    by (auto simp: image_iff algebra_simps intro!: bexI [of _ "b - a"])
haftmann@69661
   654
qed
haftmann@69661
   655
haftmann@69661
   656
lemma translation_subtract_Compl:
haftmann@69661
   657
  "(\<lambda>x. x - a) ` (- t) = - ((\<lambda>x. x - a) ` t)"
haftmann@69661
   658
  using translation_Compl [of "- a" t] by (simp cong: image_cong_simp)
haftmann@69661
   659
haftmann@69661
   660
lemma translation_diff:
haftmann@69661
   661
  "(+) a ` (s - t) = ((+) a ` s) - ((+) a ` t)"
haftmann@69661
   662
  by auto
haftmann@69661
   663
haftmann@69661
   664
lemma translation_subtract_diff:
haftmann@69661
   665
  "(\<lambda>x. x - a) ` (s - t) = ((\<lambda>x. x - a) ` s) - ((\<lambda>x. x - a) ` t)"
haftmann@69661
   666
  using translation_diff [of "- a"] by (simp cong: image_cong_simp)
haftmann@69661
   667
haftmann@69661
   668
lemma translation_Int:
haftmann@69661
   669
  "(+) a ` (s \<inter> t) = ((+) a ` s) \<inter> ((+) a ` t)"
haftmann@69661
   670
  by auto
haftmann@69661
   671
haftmann@69661
   672
lemma translation_subtract_Int:
haftmann@69661
   673
  "(\<lambda>x. x - a) ` (s \<inter> t) = ((\<lambda>x. x - a) ` s) \<inter> ((\<lambda>x. x - a) ` t)"
haftmann@69661
   674
  using translation_Int [of " -a"] by (simp cong: image_cong_simp)
haftmann@69661
   675
haftmann@69502
   676
end
haftmann@69502
   677
haftmann@41657
   678
wenzelm@63322
   679
subsection \<open>Function Updating\<close>
paulson@13585
   680
wenzelm@63322
   681
definition fun_upd :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> 'b)"
wenzelm@63324
   682
  where "fun_upd f a b = (\<lambda>x. if x = a then b else f x)"
haftmann@26147
   683
wenzelm@41229
   684
nonterminal updbinds and updbind
wenzelm@41229
   685
haftmann@26147
   686
syntax
wenzelm@63322
   687
  "_updbind" :: "'a \<Rightarrow> 'a \<Rightarrow> updbind"             ("(2_ :=/ _)")
wenzelm@63322
   688
  ""         :: "updbind \<Rightarrow> updbinds"             ("_")
wenzelm@63322
   689
  "_updbinds":: "updbind \<Rightarrow> updbinds \<Rightarrow> updbinds" ("_,/ _")
wenzelm@63322
   690
  "_Update"  :: "'a \<Rightarrow> updbinds \<Rightarrow> 'a"            ("_/'((_)')" [1000, 0] 900)
haftmann@26147
   691
haftmann@26147
   692
translations
wenzelm@63322
   693
  "_Update f (_updbinds b bs)" \<rightleftharpoons> "_Update (_Update f b) bs"
wenzelm@63322
   694
  "f(x:=y)" \<rightleftharpoons> "CONST fun_upd f x y"
haftmann@26147
   695
blanchet@55414
   696
(* Hint: to define the sum of two functions (or maps), use case_sum.
blanchet@58111
   697
         A nice infix syntax could be defined by
wenzelm@35115
   698
notation
blanchet@55414
   699
  case_sum  (infixr "'(+')"80)
haftmann@26147
   700
*)
haftmann@26147
   701
wenzelm@63322
   702
lemma fun_upd_idem_iff: "f(x:=y) = f \<longleftrightarrow> f x = y"
wenzelm@63322
   703
  unfolding fun_upd_def
wenzelm@63322
   704
  apply safe
wenzelm@63575
   705
   apply (erule subst)
wenzelm@63575
   706
   apply (rule_tac [2] ext)
wenzelm@63575
   707
   apply auto
wenzelm@63322
   708
  done
paulson@13585
   709
wenzelm@63322
   710
lemma fun_upd_idem: "f x = y \<Longrightarrow> f(x := y) = f"
wenzelm@45603
   711
  by (simp only: fun_upd_idem_iff)
paulson@13585
   712
wenzelm@45603
   713
lemma fun_upd_triv [iff]: "f(x := f x) = f"
wenzelm@45603
   714
  by (simp only: fun_upd_idem)
paulson@13585
   715
wenzelm@63322
   716
lemma fun_upd_apply [simp]: "(f(x := y)) z = (if z = x then y else f z)"
wenzelm@63322
   717
  by (simp add: fun_upd_def)
paulson@13585
   718
wenzelm@63322
   719
(* fun_upd_apply supersedes these two, but they are useful
paulson@13585
   720
   if fun_upd_apply is intentionally removed from the simpset *)
wenzelm@63322
   721
lemma fun_upd_same: "(f(x := y)) x = y"
wenzelm@63322
   722
  by simp
paulson@13585
   723
wenzelm@63322
   724
lemma fun_upd_other: "z \<noteq> x \<Longrightarrow> (f(x := y)) z = f z"
wenzelm@63322
   725
  by simp
paulson@13585
   726
wenzelm@63322
   727
lemma fun_upd_upd [simp]: "f(x := y, x := z) = f(x := z)"
wenzelm@63322
   728
  by (simp add: fun_eq_iff)
paulson@13585
   729
wenzelm@63322
   730
lemma fun_upd_twist: "a \<noteq> c \<Longrightarrow> (m(a := b))(c := d) = (m(c := d))(a := b)"
wenzelm@63322
   731
  by (rule ext) auto
wenzelm@63322
   732
wenzelm@63322
   733
lemma inj_on_fun_updI: "inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A"
wenzelm@64966
   734
  by (auto simp: inj_on_def)
nipkow@15303
   735
wenzelm@63322
   736
lemma fun_upd_image: "f(x := y) ` A = (if x \<in> A then insert y (f ` (A - {x})) else f ` A)"
wenzelm@63322
   737
  by auto
paulson@15510
   738
nipkow@31080
   739
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
huffman@44921
   740
  by auto
nipkow@31080
   741
Andreas@61630
   742
lemma fun_upd_eqD: "f(x := y) = g(x := z) \<Longrightarrow> y = z"
wenzelm@63322
   743
  by (simp add: fun_eq_iff split: if_split_asm)
wenzelm@63322
   744
haftmann@26147
   745
wenzelm@61799
   746
subsection \<open>\<open>override_on\<close>\<close>
haftmann@26147
   747
wenzelm@63322
   748
definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
wenzelm@63322
   749
  where "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
nipkow@13910
   750
nipkow@15691
   751
lemma override_on_emptyset[simp]: "override_on f g {} = f"
wenzelm@64965
   752
  by (simp add: override_on_def)
nipkow@13910
   753
wenzelm@63322
   754
lemma override_on_apply_notin[simp]: "a \<notin> A \<Longrightarrow> (override_on f g A) a = f a"
wenzelm@64965
   755
  by (simp add: override_on_def)
nipkow@13910
   756
wenzelm@63322
   757
lemma override_on_apply_in[simp]: "a \<in> A \<Longrightarrow> (override_on f g A) a = g a"
wenzelm@64965
   758
  by (simp add: override_on_def)
nipkow@13910
   759
Andreas@63561
   760
lemma override_on_insert: "override_on f g (insert x X) = (override_on f g X)(x:=g x)"
wenzelm@64965
   761
  by (simp add: override_on_def fun_eq_iff)
Andreas@63561
   762
Andreas@63561
   763
lemma override_on_insert': "override_on f g (insert x X) = (override_on (f(x:=g x)) g X)"
wenzelm@64965
   764
  by (simp add: override_on_def fun_eq_iff)
Andreas@63561
   765
haftmann@26147
   766
wenzelm@61799
   767
subsection \<open>\<open>swap\<close>\<close>
paulson@15510
   768
haftmann@56608
   769
definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
wenzelm@63322
   770
  where "swap a b f = f (a := f b, b:= f a)"
paulson@15510
   771
haftmann@56608
   772
lemma swap_apply [simp]:
haftmann@56608
   773
  "swap a b f a = f b"
haftmann@56608
   774
  "swap a b f b = f a"
haftmann@56608
   775
  "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> swap a b f c = f c"
haftmann@56608
   776
  by (simp_all add: swap_def)
haftmann@56608
   777
wenzelm@63322
   778
lemma swap_self [simp]: "swap a a f = f"
haftmann@56608
   779
  by (simp add: swap_def)
paulson@15510
   780
wenzelm@63322
   781
lemma swap_commute: "swap a b f = swap b a f"
haftmann@56608
   782
  by (simp add: fun_upd_def swap_def fun_eq_iff)
paulson@15510
   783
wenzelm@63322
   784
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
wenzelm@63575
   785
  by (rule ext) (simp add: fun_upd_def swap_def)
haftmann@56608
   786
wenzelm@63322
   787
lemma swap_comp_involutory [simp]: "swap a b \<circ> swap a b = id"
haftmann@56608
   788
  by (rule ext) simp
paulson@15510
   789
huffman@34145
   790
lemma swap_triple:
huffman@34145
   791
  assumes "a \<noteq> c" and "b \<noteq> c"
huffman@34145
   792
  shows "swap a b (swap b c (swap a b f)) = swap a c f"
nipkow@39302
   793
  using assms by (simp add: fun_eq_iff swap_def)
huffman@34145
   794
huffman@34101
   795
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
wenzelm@63322
   796
  by (rule ext) (simp add: fun_upd_def swap_def)
huffman@34101
   797
hoelzl@39076
   798
lemma swap_image_eq [simp]:
wenzelm@63322
   799
  assumes "a \<in> A" "b \<in> A"
wenzelm@63322
   800
  shows "swap a b f ` A = f ` A"
hoelzl@39076
   801
proof -
hoelzl@39076
   802
  have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
hoelzl@39076
   803
    using assms by (auto simp: image_iff swap_def)
hoelzl@39076
   804
  then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
hoelzl@39076
   805
  with subset[of f] show ?thesis by auto
hoelzl@39076
   806
qed
hoelzl@39076
   807
wenzelm@63322
   808
lemma inj_on_imp_inj_on_swap: "inj_on f A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> inj_on (swap a b f) A"
wenzelm@63322
   809
  by (auto simp add: inj_on_def swap_def)
paulson@15510
   810
paulson@15510
   811
lemma inj_on_swap_iff [simp]:
wenzelm@63322
   812
  assumes A: "a \<in> A" "b \<in> A"
wenzelm@63322
   813
  shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
hoelzl@39075
   814
proof
paulson@15510
   815
  assume "inj_on (swap a b f) A"
hoelzl@39075
   816
  with A have "inj_on (swap a b (swap a b f)) A"
hoelzl@39075
   817
    by (iprover intro: inj_on_imp_inj_on_swap)
wenzelm@63322
   818
  then show "inj_on f A" by simp
paulson@15510
   819
next
paulson@15510
   820
  assume "inj_on f A"
wenzelm@63322
   821
  with A show "inj_on (swap a b f) A"
wenzelm@63322
   822
    by (iprover intro: inj_on_imp_inj_on_swap)
paulson@15510
   823
qed
paulson@15510
   824
hoelzl@39076
   825
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
hoelzl@40702
   826
  by simp
paulson@15510
   827
hoelzl@39076
   828
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
hoelzl@40702
   829
  by simp
haftmann@21547
   830
wenzelm@63322
   831
lemma bij_betw_swap_iff [simp]: "x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
hoelzl@39076
   832
  by (auto simp: bij_betw_def)
hoelzl@39076
   833
hoelzl@39076
   834
lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
hoelzl@39076
   835
  by simp
hoelzl@39075
   836
wenzelm@36176
   837
hide_const (open) swap
haftmann@21547
   838
haftmann@56608
   839
wenzelm@60758
   840
subsection \<open>Inversion of injective functions\<close>
haftmann@31949
   841
wenzelm@63322
   842
definition the_inv_into :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"
wenzelm@63324
   843
  where "the_inv_into A f = (\<lambda>x. THE y. y \<in> A \<and> f y = x)"
wenzelm@63322
   844
wenzelm@63322
   845
lemma the_inv_into_f_f: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> the_inv_into A f (f x) = x"
wenzelm@63322
   846
  unfolding the_inv_into_def inj_on_def by blast
nipkow@32961
   847
wenzelm@63322
   848
lemma f_the_inv_into_f: "inj_on f A \<Longrightarrow> y \<in> f ` A  \<Longrightarrow> f (the_inv_into A f y) = y"
wenzelm@63322
   849
  apply (simp add: the_inv_into_def)
wenzelm@63322
   850
  apply (rule the1I2)
wenzelm@63575
   851
   apply (blast dest: inj_onD)
wenzelm@63322
   852
  apply blast
wenzelm@63322
   853
  done
nipkow@32961
   854
wenzelm@63322
   855
lemma the_inv_into_into: "inj_on f A \<Longrightarrow> x \<in> f ` A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> the_inv_into A f x \<in> B"
wenzelm@63322
   856
  apply (simp add: the_inv_into_def)
wenzelm@63322
   857
  apply (rule the1I2)
wenzelm@63575
   858
   apply (blast dest: inj_onD)
wenzelm@63322
   859
  apply blast
wenzelm@63322
   860
  done
nipkow@32961
   861
wenzelm@63322
   862
lemma the_inv_into_onto [simp]: "inj_on f A \<Longrightarrow> the_inv_into A f ` (f ` A) = A"
wenzelm@63322
   863
  by (fast intro: the_inv_into_into the_inv_into_f_f [symmetric])
nipkow@32961
   864
wenzelm@63322
   865
lemma the_inv_into_f_eq: "inj_on f A \<Longrightarrow> f x = y \<Longrightarrow> x \<in> A \<Longrightarrow> the_inv_into A f y = x"
nipkow@32961
   866
  apply (erule subst)
wenzelm@63322
   867
  apply (erule the_inv_into_f_f)
wenzelm@63322
   868
  apply assumption
nipkow@32961
   869
  done
nipkow@32961
   870
nipkow@33057
   871
lemma the_inv_into_comp:
wenzelm@63322
   872
  "inj_on f (g ` A) \<Longrightarrow> inj_on g A \<Longrightarrow> x \<in> f ` g ` A \<Longrightarrow>
wenzelm@63322
   873
    the_inv_into A (f \<circ> g) x = (the_inv_into A g \<circ> the_inv_into (g ` A) f) x"
wenzelm@63322
   874
  apply (rule the_inv_into_f_eq)
wenzelm@63322
   875
    apply (fast intro: comp_inj_on)
wenzelm@63322
   876
   apply (simp add: f_the_inv_into_f the_inv_into_into)
wenzelm@63322
   877
  apply (simp add: the_inv_into_into)
wenzelm@63322
   878
  done
nipkow@32961
   879
wenzelm@63322
   880
lemma inj_on_the_inv_into: "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
wenzelm@63322
   881
  by (auto intro: inj_onI simp: the_inv_into_f_f)
nipkow@32961
   882
wenzelm@63322
   883
lemma bij_betw_the_inv_into: "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
wenzelm@63322
   884
  by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
nipkow@32961
   885
wenzelm@63322
   886
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"
wenzelm@63322
   887
  where "the_inv f \<equiv> the_inv_into UNIV f"
berghofe@32998
   888
wenzelm@64965
   889
lemma the_inv_f_f: "the_inv f (f x) = x" if "inj f"
wenzelm@64965
   890
  using that UNIV_I by (rule the_inv_into_f_f)
berghofe@32998
   891
haftmann@44277
   892
wenzelm@60758
   893
subsection \<open>Cantor's Paradox\<close>
hoelzl@40703
   894
wenzelm@63323
   895
theorem Cantors_paradox: "\<nexists>f. f ` A = Pow A"
wenzelm@63323
   896
proof
wenzelm@63323
   897
  assume "\<exists>f. f ` A = Pow A"
wenzelm@63323
   898
  then obtain f where f: "f ` A = Pow A" ..
hoelzl@40703
   899
  let ?X = "{a \<in> A. a \<notin> f a}"
wenzelm@63323
   900
  have "?X \<in> Pow A" by blast
wenzelm@63323
   901
  then have "?X \<in> f ` A" by (simp only: f)
wenzelm@63323
   902
  then obtain x where "x \<in> A" and "f x = ?X" by blast
wenzelm@63323
   903
  then show False by blast
hoelzl@40703
   904
qed
haftmann@31949
   905
wenzelm@63322
   906
paulson@61204
   907
subsection \<open>Setup\<close>
haftmann@40969
   908
wenzelm@60758
   909
subsubsection \<open>Proof tools\<close>
haftmann@22845
   910
wenzelm@63400
   911
text \<open>Simplify terms of the form \<open>f(\<dots>,x:=y,\<dots>,x:=z,\<dots>)\<close> to \<open>f(\<dots>,x:=z,\<dots>)\<close>\<close>
haftmann@22845
   912
wenzelm@60758
   913
simproc_setup fun_upd2 ("f(v := w, x := y)") = \<open>fn _ =>
wenzelm@63322
   914
  let
wenzelm@63322
   915
    fun gen_fun_upd NONE T _ _ = NONE
wenzelm@69593
   916
      | gen_fun_upd (SOME f) T x y = SOME (Const (\<^const_name>\<open>fun_upd\<close>, T) $ f $ x $ y)
wenzelm@63322
   917
    fun dest_fun_T1 (Type (_, T :: Ts)) = T
wenzelm@69593
   918
    fun find_double (t as Const (\<^const_name>\<open>fun_upd\<close>,T) $ f $ x $ y) =
wenzelm@63322
   919
      let
wenzelm@69593
   920
        fun find (Const (\<^const_name>\<open>fun_upd\<close>,T) $ g $ v $ w) =
wenzelm@63322
   921
              if v aconv x then SOME g else gen_fun_upd (find g) T v w
wenzelm@63322
   922
          | find t = NONE
wenzelm@63322
   923
      in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
wenzelm@24017
   924
wenzelm@69593
   925
    val ss = simpset_of \<^context>
wenzelm@51717
   926
wenzelm@63322
   927
    fun proc ctxt ct =
wenzelm@63322
   928
      let
wenzelm@63322
   929
        val t = Thm.term_of ct
wenzelm@63322
   930
      in
wenzelm@63400
   931
        (case find_double t of
wenzelm@63322
   932
          (T, NONE) => NONE
wenzelm@63322
   933
        | (T, SOME rhs) =>
wenzelm@63322
   934
            SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
wenzelm@63322
   935
              (fn _ =>
wenzelm@63322
   936
                resolve_tac ctxt [eq_reflection] 1 THEN
wenzelm@63322
   937
                resolve_tac ctxt @{thms ext} 1 THEN
wenzelm@63400
   938
                simp_tac (put_simpset ss ctxt) 1)))
wenzelm@63322
   939
      end
wenzelm@63322
   940
  in proc end
wenzelm@60758
   941
\<close>
haftmann@22845
   942
haftmann@22845
   943
wenzelm@60758
   944
subsubsection \<open>Functorial structure of types\<close>
haftmann@40969
   945
wenzelm@69605
   946
ML_file \<open>Tools/functor.ML\<close>
haftmann@40969
   947
blanchet@55467
   948
functor map_fun: map_fun
haftmann@47488
   949
  by (simp_all add: fun_eq_iff)
haftmann@47488
   950
blanchet@55467
   951
functor vimage
haftmann@49739
   952
  by (simp_all add: fun_eq_iff vimage_comp)
haftmann@49739
   953
wenzelm@63322
   954
wenzelm@60758
   955
text \<open>Legacy theorem names\<close>
haftmann@49739
   956
haftmann@49739
   957
lemmas o_def = comp_def
haftmann@49739
   958
lemmas o_apply = comp_apply
haftmann@49739
   959
lemmas o_assoc = comp_assoc [symmetric]
haftmann@49739
   960
lemmas id_o = id_comp
haftmann@49739
   961
lemmas o_id = comp_id
haftmann@49739
   962
lemmas o_eq_dest = comp_eq_dest
haftmann@49739
   963
lemmas o_eq_elim = comp_eq_elim
blanchet@55066
   964
lemmas o_eq_dest_lhs = comp_eq_dest_lhs
blanchet@55066
   965
lemmas o_eq_id_dest = comp_eq_id_dest
haftmann@47488
   966
nipkow@2912
   967
end